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74-17,815 WALSH, Thomas Michael, 1930- ATTITUDE DETERMINATION OF A SPINNING SPACECRAFT THROUGH APPLICATION OF DETECTED AND IDENTIFIED STAR TRANSITS TO THE ESTIMATION OF SPACECRAFT 1 MODEL PARAMETERS. I The Ohio State University, Ph.D., 1974 Engineering, electrical
j University Microfilms, A XJERQXCompany, Ann Arbor, Michigan j
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. ATTITUDE DETERMINATION OF A SPINNING SPACECRAFT
THROUGH APPLICATION OF DETECTED AND IDENTIFIED
STAR TRANSITS TO THE ESTIMATION OF SPACECRAFT
MODEL PARAMETERS
DISSERTATION
Presented la Partial Fulfillment of the Requirements for the Degraa Doctor of Philosophy In the Graduate School of The Ohio S tate U niversity
By Thomas Michael Walsh, B.S.E.E., M.S.E.
The Ohio S tate U niversity
1974
Reading Comlttes: Approved By
Professor F. C. Weiner
Professor C. E. Warren
Professor R. B. McGhee Professor F. C. Weiner Adviser Depsrtnsnt of Electrical Engineering ACKNOWLEDGEMENTS
As is the case in most investigations of this type, the work reported here is the result of the effort of many people. The author would like to thank all of those who contributed. The author is grate ful for the guidance and advice provided by his advisor, Professor F. C.
Welmer, throughout the graduate study program and all phases of this dissertation. The author is also grateful to Professors C. E. Warren and R. B. McGhee for serving as memberB of the Reading Committee and for their help and suggestions in the preparation of this dissertation.
Acknowledgement must also be given to P rofessor E. C. Foudriat of
Marquette University for motivating this author and for his encourage ment throughout the course of this work.
The research was performed at the Langley Research Center of NASA, who provided the time, facilities, and support for the research program.
The help of the staff of the Flight Instrumentation and Analysis and
Computation Divisions is appreciated. Among them, the author would most like to thank Mr. Dwayne E. Hinton, whose assistance in all phases of th is e ffo rt was of g reat value.
Special thanka must be given to Ms. Julia Gats for her aaeiatance in the preparation of the manuscript and to Ma. Dolllne Clayton for the final typing. Finally, without the understanding and tolerance of my wife and family, completion of this graduate program would not have been possible.
ii J
VITA
July 15, 1930.... Born - Canonsburg, Pennsylvania
1953 ...... B.S.E.E., Ohio University, Athens, Ohio
1953-1955...... Research Engineer, E le c tric a l Power Laboratory Engineers Research and Development Laboratories Ft. Belvoir, Virginia
1955-1957 ...... Research Engineer, Aerospace Division, Westlng- 1958-1959 house Electric Corporation, Baltimore, Maryland
1957-1958 ...... Research Engineer, Martin - Marietta Corporation Orlando, Florida
1959-196 2 ...... Research Engineer, Goodyear Aerospace Corporation Akron, Ohio
1962 ...... M.S.E., Akron University, Akron, Ohio 1962-1974 ...... Aerospace Technologist, Langley Research Center N ational Aeronautics and Space A dm inistration Hampton, Virginia
1963-1965...... Lecturer in Electrical Engineering, University of Virginia, Charlottesville, Virginia
1969-1974...... Lecturer In Electrical Engineering, George Washington University, Washington, D.C.
1971-1974 ...... Faculty Meofcer, U niversity of Kansas, Lawrence Kansas
1974 ...... Lecturer in Electrical Engineering, Old Dominion University, Norfolk, Virginia.
i l l PUBLICATIONS
"Analysis of a S tar F ie ld Mapping Technique for Use in Determining the Attitude of a Spin Stabilized Spacecraft." NASA TN D-4637, July 1968.
"The Project Scanner Star Mapper." NASA TN D-4742, August 1968.
"Development and Application of a Star Mapping Technique to the Attitude Determination of the Spin Stabilized Project Scanner Spacecraft." Proceedings, Spacecraft Attitude Determination Symposium (Aerospace Corporation), El Segundo, California, September 30 - October 2, 1969.
"Attitude Determination of the Spin Stabilized Project Scanner Space craft." NASA TN D-4740, August 1968.
"An Investigation of Vehicle Dependent Aspects of Terminal Area ATC Operations." Proceedings, 1972 AIAA Joint Automatic Control Conference, Palo Alto, California, August 16-18, 1972.
"New Design and Operating Techniques for Improved Terminal Area Compatibility." Accepted for Publication in Proceedings, Society of Automative Engineers National Transportation Meeting, Dallas, Texas, April 30-May 2, 1974.
iv FIELDS OF STUDY
Major Field: Electrical Engineering
Studies In Control Theory. Professor F. C. Uelmer
Studies in Communication Theory. Professor C. E. Warren
Studies in Electromagnetic Field Theory. Professor R. G. Kouyoumjian
Studies In Classical Mechanics. Professor W. H. Shaffer
Studies In Mathematics. Professor H. D. Colson
v TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS l i
VITA, i l l
LIST OF TABLES lx
LIST OF ILLUSTRATIONS x
CHAPTER
1. INTRODUCTION 1
Properties Best Suited for Stellar Attitude Determination Review of C e le s tia l Sensing Techniques Problem Description Organization of the Dissertation
2. SYSTEM EQUATIONS OF ATTITUDE AND MOTION...... 14
Space Attitude Geometry Basic Reference Frames Dynamical Model and Equations of Motion Appropriateness of the Assumed Dynamical Model for an Asymmetrical Body Effects of a Residual Control Torque on the Assumed Dynamical Model
3. DEVELOPMENT OF A PARAMETER ESTIMATION PROCEDURE...... 33
4. SINGULARITIES IN THE ESTIMATOR COVARIANCE MATRIX, ESTIMATOR MODEL AND INITIAL ESTIMATES OF THE UNDETERMINED PARAMETERS...... '4 9
Observability Near Singularities for Small Cone Angles Initial Estimates
5. DEVELOPMENT OF A MODEL FOR GENERATING SIMULATED STAR | TRANSIT TIMES ...... 65
Vi CHAPTER Page
6. SIMULATION STUDIES 77
Monte Carlo Study: Expected Accuracy of Attitude Determination Procedure Baals fo r Selection of Ensemble Sample Size Attitude Determination Procedure Error Analysis Relationship of Transit Time Errors to Least Squares Functional Residuals and a Method for Assessing the Adequacy of an Estim ate
7. FLIGHT RESULTS...... 116
Spacecraft and Launch Data Star Mapper Signal and Data Processing Procedures Sumry of Signal Characteristics and Identified Stars Parameter Estimation and Pointing Direction Computation Results Relationship of Data Anomalies and Dynamical Model Inaccuracies Periodicities In the Estimation Residuals Mon-Zero Angular A cceleration and Non-Zero Control Torques Summary of Flight Results
8. RESULTS t CONCLUSIONS AMD RECOMMENDATIONS...... 160
Results Conclusions and Recommendations
APPENDIX
A. DYNAMICAL EQUATIONS OF MOTION FOR AM ARBITRARY SPACECRAFT . . 168
Case 1. - Zero Body Torques - Symmetrical Body Case 2. - Zero Body Torques and Small Asymmetry Case 3. - Symmstrlcal Body and a Small Constant Torque
B. STAR MAPPER TIME DETECTION CONCEPT 188
Principle of Operation System Description Data Proceeslng Sensitivity and Noise Consideration Desired Characteristics Telescope and Detector Characteristics Signal and Noise Characteristics Coding and Signal D etection Performance Estimates of a Specific System System Probability of Response Glossary of Sypbols for Appendix B vii APPENDIX Page
C. STAR MAPPER DESCRIPTION AND SUMMARY OF DESIGN CONSIDERATIONS...... 220
Design Considerations Design Elements Calibration and Testing Transit Time Detection
D. STAR IDENTIFICATION PROCEDURE...... 236
E. RELATIONSHIP OF THE SELECTED ESTIMATOR TO THE SEQUENTIAL LEAST SQUARES AND WEIGHTED LEAST SQUARES ESTIMATOR. . . . 247
P. RETICLE MODELING AND MEASUREMENT OF OPTICAL PARAMETERS. . 255
Reticle Reconstruction Determination of the S lit Parameter Angles
G. RELATIONSHIP OF ATTITUDE DETERMINATION ERRORS TO ERRORS IN THE SYSTEM PARAMETER ESTIMATES...... 266
REFERENCES...... 277
vii i LIST OF TABLES
Table■ Page
1. The Sequence of RoteClona and the Trenaformatlon Matrices for the Ten Reference Frames Used to Determine Spacecraft Attitude ...... 18 2* Simulation Studies Input Data ...... 86 3. Simulation Data Liat Sample ...... 87 A. Star Catalog Input Data For Simulated Star Transits ...... 88 5. Simulated Star Transits ...... 91 6. Standard Deviation of Pointing Direction Error of i„ Axis for 10, 15, 30 and 5A Error Sequences ...... 93 7. Statistics of the Errors in X (Degrees)...... 99 8. Selected Star Groupings for the Error Study ...... 103 9. Attitude Determination Accuracy for the Case of Poorly Placed S ta rs ...... I l l 10. Scanner S p a c e c r a f t ...... 117 11. Spacecraft Moments of Inertia ...... 119 12. Control System and Data Interval Schedule ...... 120 13. Id e n tifie d S tars ...... 137 1A. Stars Used in Parameter Estimation Procedure ...... 139 15. A P rio ri Data ...... 1A0 16. Vehicle Motion Parameters for the Second Data Period of Flight One ...... 1A3 17. Control System Specifications ...... 155 18. Summary o f F lig h t R e su lts ...... 158 19. Elements o f the Right Ascension Error Term, cos bAa ...... 271 20. Elements o f the D eclination E rror Term, A b ...... 272
ix LIST OF ILLUSTRATIONS
Figure Page
1. Simplified Representation of Star Happing ...... 8 2. Basic Reference Frames of Spscecraft Attitude Geometry . . . 17 3. Orientation of Slit Plane with Respect to the Star Mapper ig - Sg F l a n e ...... 22 4. Relationship of Vehicle Principal Axes to Inertial Reference Frame ...... 23 5. Relationship of the Telescope Mounting Angle, r, to the Elevation Angle, n ...... 73 6. Geometrical Relationship of the Transit Times of the Vertical and Slanted Slit Planes ...... 73 7. Flow Diagram of E rror Time H istory Generation ...... 80 8. Relative Aslmuthal Positions of Stars Transited in Simulation ...... 89 9. 99X Confidence Bounds on Standard Deviation of ig Axis Pointing Direction Error (Star 48 - Worst Case) ...... 96 10. 99% Confidence Bounds on Standard Deviation of ig Axis Pointing Direction Error (Star 154 - Worst Case) ...... 97 11. 99X Confidence Bounds on Standard Deviation of ig Axis Pointing Direction Error (Star 239 - Worst Case) ...... 98 12. Maximum Standard Deviation in Right Ascension of ig Axis vs. Number of Transited Stars Per Spin Period ...... 104 13. Maximum Standard Deviation in Declination of ig Axis vs. Number of Transited Stars Per Spin Period ...... 105 14. Har1mi«AStandard Deviation of Pointing Direction Error for lfi Axis vs. Nuabsr of Transited Stars Per Spin P erlo a ...... 106 15. Maxlaum Standard Deviation of Pointing Direction Error, o _ (tj)* g , w ith 99X Confidence Bounds vs. Number of S tar T ra n s its Far Spin Period (1/2 Precession Period) . . 108 16. Maximum Standard Deviation of Pointing Direction Error, Op(ti)£s» with 99X Confidence Bounds vs. Number of Star Transits Per 9pin Period (1 Precession Period. . . . 109
-x. Figure Page
17. Maxima Standard Deviation of Pointing Direction Error, °p(ti>18> wlth 991 Confidence Bounds vs. Number of Star Transits Per Spin Period (1-1/2 Precession Period) . 110 18. Nominal and + 3o T rajectory P ro file s ...... 11B 19. Illustration of Control Action of the Attitude Control System...... 122 20. Spacecraft Operational Sketch ...... 124 21. Illustration of Reticle and Pulse Groups Generated by S tars 1 and 2 ...... 125 22. S tar Mapper Data Processing Procedure...... 127 23. Typical Scan o f S tar Mapper F ield of View ...... 131 . 24. Oscillograph Record of Output of Star Mapper ...... 136 25. Pointing Direction of the Star Mapper Optical Axis (1») for One Vehicle Spin Period Based on the Set of Nine Vehicle Motion Parameters Selected from the Second Data Period of Project Scanner Flight 1 for Time Beginning 6h 25m 46.78 UT on August 16, 1966 ...... 145 26. Pointing Direction of Vehicle Principal Spin Axis kg for One Precession Cycle Based on the Set of Nine Vehicle Motion Parameters Selected from the Second Data Period of Project Scanner Flight 1 for Time Beginning 6h 25m 46.7a UT on August 16, 1966 ...... 146 27. Pointing Direction of ko Axis for One Precession Cycle Based on the Set of Nine Vehicle Motion Perameters Selected from the Second Data Period of Project Scan ner Flight 1 for Time Beginning 6h 25m 46.7s on August 16, 1966 ...... 147 28. Increase of Total Angular Spin Rate During the Second Data Period of Scanner Flight 1 ...... 153 29. Illustration of the Source of an Acceleration Torque due to a Mlsallnad Jet ...... 154 30. Simplified Drawing of Star Mapper...... 189 31. Simplified Block Diagram of System ...... 191 « 32. Incoming Signal and Computed S tar Map ...... 193 33. Data Processing Block Diagram ...... 194 34. Spatial Density of Stars ...... 196 35. Spatial Density of Stars per Revolution of a 6° Vertical Field of View ...... 197 ,36. Block Diagram of Detection Process ...... 202
51 Figure Page
37. Assumed Blectron Distribution for Noise-Only and Signal- Plus-Nolse Samples ...... 205 36. Probability of Detection as a Function of Visual Magni tude...... 211 39. Frequancy of System Response as a Function of Visual Magnitude, with a Background of 160 Tenth-Magnitude Stars par Square Degree ...... 212 40. Frequancy of System Response as a Function of Visual Magnitude, with a Background of 500 Tenth-Magnitude Stars per Square Degree ...... 213 41. Frequancy of System Response as a Function of Visual Magnitude, with a Background of 1000 Tenth-Magnitude Stars per Square Degree ...... 214 42* System Figure of Merit Versus Code Length...... 216 43. S tar Mapper Concept ...... 221 44. Simplified Signal Flow Diagram ...... 223 45. Measured Spectral Response of the Star Mapper ...... 228 46. Reticle Configuration ...... 229 47. S tar Mapper C alib ratio n ...... 232 48. Decoding of Star Mapper Signals ...... 234 49. Simplified Scan of the Star Mapper Extended Field of View About the Celestial Sphere for the Case of Zero Cone A n g le ...... 238 50. Angular Separation of Two Stars on the Caleatlal Sphere. . . 240 51. Reticle Reconstruction Teat Setup ...... 256 52. Schematic of Theoretical R eticle ...... 258 53. Reconatructed Reticle Outline ...... 259 54. Geometry of Reticle Measurements for the Determination of ...... 261 55. Reticle Rotation Due to Mlaallnement Angle, ...... 264 56. Right Ascension and Declination of ig and kg . ; ...... 267 P late
I. Typical Coded Star Signals from Alkald and Alphecca During the First F light ...... 132
xii CHAPTER 1
INTRODUCTION
In recent years, spin 'stabilised rockets and spacecraft have been used to conduct research in meteorology and earth related features.
Other spacecraft with spin stabilised elements are being considered for interstellar x-ray measurements, for earth resource research, and for the study of local and general atmospheric pollution. A common requirement In these experiments is the time history reconstruction of the llne-of-slght pointing direction of onboard research Instruments.
This feature is necessary to relate the research Instrument data time history to spacecraft spatial time history in order to locate and identify the source radiation and its characteristics. In addition,
Increasing demands have been placed upon the reliability of instrumen tation systems for spacecraft operations. In this connection, a question arises as to whether high levels of reliability and accuracy in attitude determination systems can be achieved through use of relatively simple measurement techniques with sophistication of elabor ate Instrumentation being replaced by more sophisticated data processing techniques. This question has been examined in this dissertation, especially in terms of instruments having no moving parts. The general approach taken in this study 1 b to investigate the use of a celestial reference frame for attitude determination measurements.
1 Properties Beat Suited for Stellar Attitude Determination
Men have used star patterns for navigation for centuries. How ever, It Is difficult to achieve automated instrumentation on the same level of sophistication as the decision processes found in man* It Is desirable therefore to find ways for simplifying the problem, so that it becomes a feasible task for the state-of-the-art Instrumentation.
It must be realized that the simplification procedures must preserve the uniqueness features of stars; otherwise, the celestial attitude information obtained becomes plagued with uncertainties.
The known information about stars has been arranged into star catalogs and consists of position, spectral intensity and magnitude classification. Star magnitude is a complex quantity, since it depends upon the net energy sensed by a particular detection system from a particular star. Usually, apparent visual magnitude (mv> is used by navigators to assess grossly the brightness of a star. For space application there is considerable uncertainty about the actual spectral content of star radiation outside the earth's atmosphere. The present- day knowledge of star radiation spectral content is restricted to the atmospheric window regions and has been Incompletely explored even in this limited region.
The star positions are recorded in a star catalog more accurately than either the star magnitude or the spectral type, because angular measurements are more easily made than radiation measurements. It should be noted further that the accuracy requirements of star measure ments for the purpose of navigation are not necessarily the same as the measurement accuracy required for attitude determination. Both 3 requirements are very much dependent upon the mission. However, since the task of star Identification must be accomplished regardless of the nature of the mission, the measurement accuracy requirements depend upon the allowable ambiguity of the identification. The main effect of measurement accuracies upon star identification 1 b the inability to distinguish between small differences of selected stellar properties from one star to another. Star positions are specified in a star catalog in spherical coordinates, with the dimension of the radial distance equal to one (ref. 1]. If the angle between two stars can be measured to an accuracy of one arc minute and with an assumed full range of 0* to 360* for the angle, the uncertainty in angle measure ment is approximately 0.005Z. This value shows the discrimination potential of angle comparison for star identification. The philosophy of this dissertation Is to use the grouping characteristics of the stars as a unique identification feature and the angular separation of the stars as a measurement parameter. This uniqueness is the feature which has long been used by man in the identification of the constellations.
It is the Intent of this study to seek a concept whereby star patterns can be used as a basis for the attitude determination problem.
The chosen star groupings should have a very simple geometry so that time-consuming computations in Identifying it are avoided. On # the other hand, it is desirable to perform the identification uniquely, and the simplest approach is to employ a more complex pattern. The human Id e n tifie r has managed to do so by dividing s ta rs in to co nstel lations, using crudely measured relative position and gross relative « determination of brightness. This 1 b a good example that the use of more stars for uniqueness of pattern identification la a workable approach. The simplest star pattern w ill consist of three stars, since even if the relative position of two stars is unique, their position on the celestial sphere will have a 180° ambiguity. There fore, this study will consider three-star patterns as a simplest configuration.
Review off Celestial Sensing Techniques
Early work on celestial sensing devices involved the study and development of star tracking systems. The possibilities of celestial sensing without closed-loop tracking were not extensively considered.
Probably the significant exception to this was the work done with image
tubes in which a gimbailed optical system was approximately pointed at
the target and the final measurement was made by the image tube.
Subsequently, a number of investigators who were interested in
the general problem of attitude determination in space considered using
image tubes with wide angle optical systems. With this type of system, a sufficient number of bright stars could be detected to achieve auto matic pattern recognition for a random orientation. A system of this
« type was suggested by Rosenfald [ref. 2]. Employing a field of view of about ten degrees, he achieved an accuracy of a few minutes of arc
and detected stars down to the sixth magnitude. Another system dea-
* cribed by Potter [ref, 3] employed a field of view of 30 degrees, achieved an accuracy of approximately seven minutes of arc, and de
tected stars down to the third magnitude. Both of these systems had
the decided advantage of requiring no closed-loop tracking, but were
somewhat lacking in either accuracy or in requiring the detection of
very faint stars. More recently, efforts have been made to develop mosaic or grid type celestial seniors, which would avoid the need for an image tube, and which employ no moving parts. Systems of this type have not provided adequate resolution to be competitive with star trackers while providing a sufficiently large field of view. However, with
the Improvement of solid state materials, future sensors may be of
th is type.
A partial solution to the problem of attaining a high resolution has been achieved by a novel device described by Snowman [ref. A] in which a highly accurate attitude measurement (30 arc seconds) was achieved for all three axes with a 46 degree field of view. In this
case, various reference star fields were mechanically fabricated and mounted at the focal plane of the optical system. This device re quired, however, that it be pointed within ten degrees of the center of the reference field, and the problem of randomly pointing the sensor
In any direction relative to the celestial sphere was not solved.
A study of the various system trade-offs led Lillestrand and
Carroll [ref. 5] to conclude that wide-fleld-of-view systems offer considerable promise, If the problem of achieving a sufficiently high
resolution can be solved. By employing a narrow optical s lit to Bean
the star field, the position of the images can be found to an accuracy of at least 1/10,000 of the field of view of the optical syBtem. This means that optical systems with a 30 degree field of view can provide
an accuracy of ten seconds of arc, as described by Harrington [ref. 6].
In the case of Inertlally* stabilized spacecraft, provision must be made
for rotating the slit. In the case of spinning spacecraft, Kenlmer and Walsh [ref. 7] conjectured that systems of this type having signi ficantly smaller fields-of-view could be fabricated with no moving parts. For both cases, a narrow slit is mounted at the focal surface of the sensor for sensing a transit of a star across the sensor's field of view. This slit can permit the accuracy of attitude measure ments to approach the optical resolution of the sensor.
I f the frame to which the s ta r sensor is mounted undergoes random changes in orientation as a function of time, then it is not feasible
to compensate analytically for this random motion. However, if the orientation change is systematic, then it should be possible to account
for such changes. In this dissertation, it is assumed that the scan ning sensor is rigidly attached to a spinning body. Under this assumption, a technique for analytical compensation for the vehicle's
orientation changes w ill be Investigated.
Problem Description
Stellar attidue determination.is defined here to consist of three
major sub-problems: detection of stars falling within the field of view of the stellar sensor; identification of the detected stars through
correlation of their angular separations with cataloged values; and
estimation of the parameters defining the dynamical equations of motion
and spacecraft and sensor geometry through use of this stellar transit
time history. Successful estimation of these parameters will result in
the necessary quantities required for computation of the spacecraft
attitude between transits. The star detection and identification pro
blems are described in the main text and solutions for these problems
are developed in an appendix. The major problem to be addressed In this dissertation is that
of determining the attitude of a spinning spacecraft through applica
tion of estimation techniques, given the dynamical equation model of
spacecraft motion, and the times of sightings of stars as observed in
a sensor reference frame attached to the spacecraft. The vehicle to be considered is characterized as a spin stabilized, torque free,
rigid body. With a passive optical system on board, vehicle motion
results in a scan of the celestial sphere. The detection and identi
fication of stars falling with this instrument's field of view result
in a star map that is descriptive of spacecraft attitude relative to
the celestial sphere. A simplified representation of star mapping is
illustrated In Figure 1. It Is clear that for this simple case,
Instantaneous spacecraft attitude can be determined by correlation of
the generated star map with known stellar positions.
The dissertation covers> consideration of the design aspects of the star telescope, including a discussion of advantages of the application of optical coding techniques to improve the detection of star signals by enhancement of discrimination of stars brighter than a limiting visual magnitude against the background of dltmier stars. A technique of Identifying the detected stars is discussed. Procedures for determining the attitude time history of selected spacecraft* axes using the sighting times of the identified stars are developed. These procedures Involve the application of minimum variance techniques to the determ ination of unknown, or poorly known, parameters in the space craft dynamical equations of motion or geometry.
The accuracy of this attitude determination procedure Is lnvestl- i gated by computer simulation studlea. In this simulation an exact 8
Spacecraft spin
Instantaneous field of view
Figure 1. Simplified Representation of Star Mapping. 9 spacecraft dynamical model and statistically Independent star sighting time errors are assumed. A Monte Carlo study Is used to determine the sensitivity of the attitude determination procedure to such variables as sighting time errors• star placement, and nunfeer of sighted stars.
The result of this simulation effort is a prediction of the expected accuracy of the attitude determination procedure. The simulation Is also used to Investigate the feasibility of establishing criteria to be used as measures of "goodness" of estimates of the dynamical equation parameters In a real world situation.
To illustrate the usefulness of the discussed attitude determina
tion procedure, this estimation technique Is applied to flight data obtained from two sub-orbital probes launched from Wallops Island,
Virginia [refa. 8, 9]. The procedure was Implemented using ground baaed data processing of stellar time transits telemetered to a ground station. The flight vehicles used for these launchings may be very closely approximated by symmetrical, torque free, rigid bodies for the
flight period of interest. Effects of spacecraft model Inaccuracies
(Including assynmetrles, varying moments of Inertia and constant
torques) on the attitude determination results are discussed.
Organisation of the Dissertation
In addition to the background material and problem definition of
the Introduction, the dissertation is divided into five main chapters.
Chapter 2 deals with a description of the problem geometry, assumed
spacecraft models, and a formulation of the dynamical equations of motion for the assumed models. This discussion includes a development
of the relationships of the dynamics of the basic reference frames for 10 the model and the orientation of these reference frames to the coordi nates of the celestial sphere. A more detailed formulation of the dynamical equations of motion may be found In Appendix A.
The third chapter contains a development of a vehicle motion parameter estimation procedure based on the observation time of sighted stars. This chapter presents a general formulation of the weighted least squares method used In the estimation. Also Included In the third chapter Is a development of a relationship between transit time measure ment errors and residuals of the estimation procedure.
Chapter 4 deals with a discussion of observability of the selected system states, conditions of singularity of near singularities of the estimator covariance matrix, and techniques for computation of estimates used to initialize the estimator process. The relationship of the selected least squares estimator algorithms to the Kalman estimator algorithms Is presented In Appendix E.
A significant portion of the study of this dissertation deals with simulation studies, particularly as related to accuracy of the results of the parameter estimates and resulting accuracy of attitude determination using these parameter estimates. The material presented
In Chapter 5 describes the development of a model and simulation tech nique for the generation of star transit data. A procedure for star identification, using the assumed spacecraft dynamical model and star transit time data, Is described in Appendix D. Procedures for processing of flight data were developed through simulation studies which made use of the parameter estimation program, the star transit time generation program, and the star Identification program. The sixth chapter deals with the principal simulation studies of 11 this dissertation* The spacecraft model used In the study was selected
to agree closely with Che model assumed for a flight vehicle. The
dynamical model used to generate transit time data Included an asym
metrical body while the dynamical model of the estimator was that of
a symmetrical body. The simulated transit time data were adulterated
with time errors prior to use of the estimation procedure. These time
errors were defined to be random with statistical properties specified
by the star sighting Instrument and expected background light discussed
In Appendixes B and C. All errors in parameter estimation are related
to errors In the resulting attitude determination computations for
se lec te d axes In e q u a to ria l coordinates. A Monte Carlo study was con
ducted to determine the effects of measurement noise, star sighting
frequency, angular placement of observed stars, and data processing
period in terms of fractional parts of a precession cycle on parameter
estimation results and subsequent attitude determination computations.
Additional error studies were conducted to determine the sensitivity
of pointing direction errors to state parameter estimate errors. The results of these error sensitivity studies are summarized In Appendix G.
Chapter 6 also includes a discussion of the relationship between
residuals of the parameter estimation procedure and transit time measure
ment errors. This relationship was used as a basis for a method,of
assessing the adequacy of the solution of a set of state estimates
for any data processing period. This latter development was necessary,
especially for flight data processing, since no other attitude reference
was available to check the results of the attitude determination
procedure. 12 Scar sighting Instruments of the type described In this disserta tion were flown on two sub-orbital flights. A brief description of the nominal trajectories, spacecraft, payload and attitude control system is given In Chapter 7. A brief description of the star mapper and data processing Is also presented. More detailed descriptions of the tran sit time detection technique and star mapper design characteristics may be found in Appendixes B and C. Flight results are summarized In terms of detected and identified stars of the two flights. Signal and noise characteristics observed in the flight data are compared to predicated characteristics. Anomalies noted In the flight results are discussed and related to omissions in the assumed spacecraft dynamical model.
Estimates of the spacecraft model parameters obtained from processing of the identified stars and their transit times are presented. Examples of pointing direction computations for selected axes resulting from these estimates are discussed. An estimate of the accuracy of these attitude or pointing direction computations is determined on the basis of estimates of the statistics of transit time measurement errors experienced in the two flights.
The results presented here demonstrate the feasibility of obtaining extremely accurate estimates (e.g. 0.006°) of attitude determination through the use of relatively simple instrumentation coupled with, com prehensive detection, identification and estimation procedures. Opera tional systems for long term missions will require refinements such as additional observables (i.e. disturbance torques and a sun sensor for continuous daylight operation) and compensation for systematic errors in the assumed model due either to negligent omission or to long term dynamical changes. Recent requirements established by NASA's program 13 of Advanced Applications Flight Experiments point out a; goal of 0.001° of accuracy for attitude determination of satellites engaged in earth resources surveys and geodesy experiments. This Increase In accuracy over that obtained in this dissertation point out a need for further study, especially as related to satellite applications. It should be noted that this accuracy goal of 0.001° approaches the nominal accuracy of the .cataloged positions of stars. Therefore, greater attention must be given to more exact modeling of spacecraft dynamics and problem geometry In order to reduce the error contributed by uncertainties in known stellar positions. In addition to this new accuracy goal for earth applications a recent proposal has been made to apply the techni que of this dissertation to the problem of detecting and tracking asteroids [ref. 10]. CHAPTER 2
SYSTEM EQUATIONS OF ATTITUDE AND MOTION
The two primary operations necessary fo r the use of s ta r mapper information for attitude determination are: the Identification of detected stars* and the estimation of the parameters in the dynamical equations of motion which describe spacecraft attitude as a function of time. Before discussing the development of star identification and parameter estimation procedures* a definitive description of the geometry and equations of the general attitude problem for an assumed dynamical system is necessary. In this chapter the governing differ ential equations for an assumed mathematical model are presented.
Space A ttitu d e Geometry
The problem considered here is that of a geometrical formulation of the celestial attitude of a selected reference frame fixed in a spinning, torque-free, prolate, symmetrical, rigid body. A more general discussion of the dynamics of a spinning rigid body is presented in
Appendix A in which the three special cases, (1) a prolate, symmetrical, torque-free body; (2) a prolate, slightly asymmetrical, torque-free body; and (3) a prolate, symmetrical body subjected to a single constant torque, are considered. The appropriateness of the dynamical model of the first case for small asymmetries will be discussed in this chapter. The effects of a small constant torque on the dynamics of the first case will be discussed here and also at greater length in a later chapter 14 15 describing flight results.
The general problem of this study Is defined as one for which the orientation of the body-fixed reference frame Is to be determined using the spacecraft dynamical model with an estimation of the parameters which uniquely define Its orientation. The basic or fundamental measurements to be used in the estimation of vehicle orientation are the times at which known stars transit specific slits of the star mapper reticle.
It should be emphasized that the vehicle Is assumed to be operating In an environment In which the torques due to aerodynamics, gravity gra dient, and solar radiation pressure produce only long term effects and can be neglected for the short term attitude determination problem.
During the periods of time in which stars are observed and the attitude of the spacecraft body Is estimated! the body's angular momentum vector is assumed to have attained a given orientation rela tive to a specified Inertial reference frame. The spacecraft reference axes describe coning and nutating relative to the angular momentum vector. It Is further assumed that the angular momentum vector is periodically reoriented relative to this inertial reference frame through use of a reaction jet control system. Although this reorienta tion control problem is not the subject of this paper, It is considered appropriate to mention significant results of studies directed toward this problem. Some results have been obtained for the fuel-optimal
[ref. 11, 12] and time-optimal [ref. 13, 14] nutation reduction problem and studies made of the related time-optimal reorientation problem
[ref. 15]. In addition, .theoretical fuel-optimal control laws have been analysed for the combined task of nutation reduction and angular momentum vector reorientation [ref. 16]. For the spacecraft considered 16 in this study, both attitude and rate control are achieved by a single body-fixed reaction Jet thruster which produces a torque about the center of mass and causes a gyroscopic precession of the spin axis.
The desired control Is attained by varying the timing and time duration, of the thrust. It Is assumed that the mass of the control fuel Is small in comparison to the overall mass of the vehicle, and thus, the mass and Inertia properties of the system remain constant throughout the control and stellar observation periods. Further, it is assumed that the jet turns off and on instantly and effects of valve hysteresis and time lags are negligible. Finally, the jet does not affect the vehicle dynamics during the stellar observation periods.
Basic Reference Frames
The geometrical model for this problem Involves the use of eight basic reference frames to define the principal axes with respect to the celestial sphere. Tvo additional reference frames allow for star mapper mounting mlsalinements or uncertainties in the angular relationship of the vehicle principal axes to known vehicle body axes. The reference frameB assumed fo r th is problem are shown In Figure 2. The sequence of rotations and subsequent transformation matrices for the defined Euler angles are listed in Table 1. The selected reference frames and their re la tio n sh ip s shown in Figure 2 are:
1. Celestial reference frame with unit vectors 1^, j^, k^. The
A A unit vectors 1^ and k^ are selected to be in the direction of the first point of Arles and the North celestial pole, respectively. The unit vector j^ is arbitrarily selected to lie in the celestial equatorial plane to complete a right hand system. 17
a. Relationship of the celestial b. Relationship of the inertial reference frame to the Inertial reference frame to the vehicle reference frame. principal axes. A
10
A A i7.is
c. Misalinement of the vehicle d. Relationship of the star principal axis relative to the mapper reference frame to star mapper reference frame. any s l i t plane reference frame.
Figure 2. Basic Reference Frames of Spacecraft Attitude Geometry. 18 TABLE 1 THE SEQUENCE OF ROTATIONS AND THE TRANSFORMATION MATRICES FOR THE TEN REFERENCE FRAMES USED TO DETERMINE SPACECRAFT ATTITUDE
r « i * n " c o s * - s i n * 0 ‘ H h
A A • m - ( ♦ ) $ ( * ) s i n * c o s * 0 J 1 h
A A L 0 0 1 . ■ k 2 ■
A H ‘ 1 0 0 * * 2 * 3
• -(e) » ( 0 ) - 0 C O S 0 - s i n e ^ 2 h A
“a ■ r a «i "cos^ -sln$ 0 ■ i 3 *« A • - ( + ) * ( * ) m 8in A A 0 0 1 - k 4 A .k 3 . A W A ™ P A •! ' 1 0 0 ' ‘♦I * s A -(e) t ( e ) - 0 c o s e - s i n e 3 * A . 0 s l n B c o s ^ £ a k 5 u ^ « A A r * ■ r a n ’ c o s if i - s i n i j i 0 * ^ 5 V A A • - ( * > t ( * ) M s i n ^ c o s j|> 0 J 5 h A A . 0 0 1 . , k 5 . “ a “ m A " ■ * 1 0 0 h « A A • 9 - 0 cos - s i n J 6 - . 0 s i n c o s . k 6 . . k 7 _ e l - TABLE 1.—Continued. ■A A cos e, 0 s in e, 7 i 8 A A " ( e 2 ) (e2 ) 0 1 0 h J 8 A A -sin e, 0 COS c Lm k7 * <■J - k 8 . 7 r ■ A A A cosy -sln y 0 *8 H A A 38 - ( Y ) J g ( Y ) - Blny cosy 0 A k. 0 0 1 • ' A A 1 0 0 9 i 1 0 A A J g - ( 0 ) ( 0 ) 0 COS0 -sinB 3 1 0 A 0 sinB cosB k 1 0 J A • 20 2. Inertial reference frame defined by two Euler angles © and$ relative to the celestial reference frame. In this frame the unit A vector kg is defined as parallel to the angular momentum vector of the spacecraft. 3. A body fixed reference frame defined by the Euler angles 6 and ^ relative to the inertial reference frame* In this frame the ^ a a unit vectors ig, jg, and kg are defined to be parallel to the princi- A pal axes of the body. The unit vector kg is defined to be along the vehicle spin axis. Since the vehicle is assumed to be symmetrical, the unit vectors £g and jg may be selected to be any mutually orthogonal A pair which are also orthogonal to kg. For convenience the lg vector Is considered to be directed along the nominal optical axis of the star mapper. A. The reference frame of the star mapper relative to the space craft principal axes is defined by two mlsallnement Euler angles and £2 * These two angles allow for mlsallnement of the star mapper system relative to the spacecraft principal spin axis as measured prior to launch and also allow for changes In the location of the spin axis A after launch and rocket fuel burnout. In this reference frame the lg vector is selected to be along the optical axis of the star mapper. 5. The last reference frame of interest is the frame defining the location of a slit plane (vertical or slanted) relative to the star mapper optical reference frame, TVo Euler angles, y and 6 , define this frame relative to the optical frame. There are actually two sets of y and B» each set defining the vertical (yv, 8^ and slanted (yQ, Ba) slit planes respectively. In either case the I^ q and u n it vectors are defined to be in the vertical or slanted alit plane. 21 ~ One additional angle le needed to complete the geometrical formu lation of the problem. This angle defines the coordinates of a star as sighted in a slit plane reference frame. Figure 3 Is a representa tion of a slit plane and the angles y and 0 relative to the optical axis of the star mapper. In this figure a star sighting is shown In A the slit plane at an angle r\ relative to the intersection of the ig - Jg and I^q - planes. A unit vector to a star sighted in the slit plane reference frame is ^ A A £ - cosn i 10 + 0 J 10 + sinn k 10 . (2, 1) Figure 4 illustrates the geometry of the problem with certain exceptions For simplicity the angles c ., e_, y , 0 , y , 0 , n * and n are not A Z V V 0 8 V 0 shown in th is fig u re. The angles y and 0 are reticle configuration parameters and must be determined prior to a flight. A discussion of a technique which may be used to experimentally determine the values of these angles is given in Appendix F of this paper. For the purpose of the present dis cussion it is sufficient to state that the angles 0 and 0 are defined v s as 8 - 0 and 0 ■ 0. As stated previously, pairs of transit time V 8 produced by a single star represent the star's transit of two sets of points defined on the reticle. Schematically, these points lie on each of two analytical lines located on the reticle by selected value? of y , y , 0 and 0 . These two analytical lines are designated here as V 8 V 8 the vertical slit plane and the slanted slit plane. The relationship of the lg axis or the optical axis of the star mapper, with zero misline ment angles and e^» to these two slit planes is illustrated in Figure 4. A more detailed description of the relationship of these slit planes to the vertical and slanted coded optical gratings of the 22 Figure 3. Orientation of Slit-Plane with Respect to the Star Mapper ig “ Plane. 1 23 Principal Vertical Axis of Slit Body r - Slanted Slit Scan Plane of Optical Axle i t Figure 4. Relationship of Vehicle Principal Axes to i Inertial Reference Frame. 24 reticle is given in Appendixes C and F. Dynamical Model and Equations of Motion The relationship between the coordinates of an identified star In the celestial reference frame and a slit plane (vertical or slanted) reference frame may be shown, using the defined Euler angles, as R « (*) ( 0 ) (*) ( 6 ) (*) (Gl) (e2) (y) ( 8) r (2 . 2) where jr has been defined as A A A r - cosn i 10 + 0 Jio + 8lnn kio • Using standard notation for right ascension (a) and declination ( 6 ) , the vector R as illustrated in Figure 3, may also be written as A A R ■ coao cosfi i^ + sin a c o b 6 + sinfi . (2.3) The derivation of the dynamical equations associated with ♦ , ©, $, 0 , and of equation 2.2 are presented in Appendix A. It should be noted that this derivation allows for non-symmetry and also permits control torques to be applied to the system. For the assumed model of this study, that Is, symmetry and no torques, the dynamical equations are where h Is Che magnitude of the system angular momentum vector expressed In the Inertial reference frame (S^). The solutions for 4>, V* h are (2.5) ( 2. 6 ) (2 .7 ) (2.8) V I COS0 . (I - I, to * ----- (2 .9 ) Here 1^ is the moment of inertia about the principal spin axis of the vehicle; I is the moment of inertia about either one of a pair of axes mutually orthogonal and orthogonal to the principal spin axis of the body; and is the total spin velocity about the principal spin axis* If all parameters of the transformation in equation 2.2 were uniquely defined then the attitude of any set of axes would be known. However, only the y and 0 angles are well known before flight. Of the remaining parameters in equation 2. 2: and e^ are assumed to be unknown; in it i a l estimates of • and 9 are assumed to be available from launch data; an initial estimate of was available from preliminary examination of * * star mapper data; only rough initial estimates of 4) and are available sin ce 6 is assumed to be unknown; and and \|tQ are unknown. A develop ment of a means of estimating these unknown or poorly known parameters through use of the identified stars and their observed times of sightings will be the major effort of this dissertation. The development of the 26 estimation technique is discussed in Chapter 3 and a means of detecting sighting times and a technique of Btar identification w ill be discussed in Appendix D and also In a later chapter on flight results. Appropriateness of the Assumed Dynamical Model fo r an Asymmetrical Body The development of the dynamical equations of motion thus far has used the assumption that the moments of inertia, and ^ were equal and the body was subjected to rero torques. The errors produced by the equal moments of inertia assumption w ill be examined here for the case of a small asymmetry and small cone angle. From Appendix A, a spinning, torque-free, rigid body, with two slightly unequal momentB of I n e r tia , moves such th at 0 ■ 0O- “ ■§■ a (tan6 o )(l+r)[cos 2(it + o■§■ (sec20 o ) (l+r)(cos4pt - (2 .10) 2 cos 6 o )] + .... $ - \|»o + Jit + -| (l+r)(2+tan 2eo) ( s in 2ut)[l + -| (1+r) (2.11) x ( 2+tan 20o) ( 2+cospt - (■ 1+COB 00 $ ■ n - Y [1 +. c2(l - “ (l+r) (2 + tan 20o) ) ] + ...... It should be noted that for c ■ 0, equations 2.10, 2.11, and 2.12 degenerate to the expressions of G, and $ as defined by equations 2.4 through 2.9 for the Byanetrical caae. how consider a reference frame {Sg8> bo as to satisfy equations 2.10, 2.11, and 2.12 with e ■ 0 (Ij^Ij). This is the same system assumed In this study. Next consider another reference frame (S^} which moves with respect to {S^J so as to satisfy equations 2 . 10, 2. 11, and 2.12 w ith e 0. The problem now 1 b to examine the misorientatlon of {S, } with ■ oa respect to {Sga> as a function of time. In general, there always exists an axis such that a single rotation about this axis will cause {Sga) to coincide with (Sgg) [ref. 17]. Let this rotation be represented by c* Then it can be shown that cosC/2 - cosA0/2 cos -A&yA* (2.13) where AO, Ai|> and A^ represent the portions of equations 2.10, 2.11, and 2.12 due to e j 0. Equation 2.13 may be written as (l-ain2C/2)1/2 - (l-sin2A0/2) V 2(l-sin2 or ( l- s in 2c/2) - l-ain 2A0/2 - ain 2 M - t M + 8in 2 M sln2 MjL4£ a ln 2c/2 - a in 2A0/2 ( l- a in 2 --^-y -At.) + a in 2 s in 2£/2 • a in 2A0/2 coa 2 ^ ^ + gin 2 At jLAft , (2.14) For small A0 and Aty + At equation 2.14 becomes (C/2) 2 - (A0/2) 2 + Aft)2 or C - (A0) 2 + (A* + A* )2 (2.15) 2 2 2 Collecting the terma (A0) and (At|t + A$) and neglecting terms 0(e ) 2 and 0(0 ) and higher order ve have o c - | eo (l+r) C0. 2HC . (2 X6) The maximum value of this mlsallnement angle C w ill be '« « - I °o <1+r> • • ' For the study of this dissertation, this angle C will be of max negligible value. The vehicle ia assumed to be of a pronounced pencil * shape, I » I3; in fact I« 20* 3 * Then l + r w ill be approximately -0.05. The cone angle 0o w ill be held to an angle of leas than 0.5 degrees and 1^ and will be matched to within + 10JC, and therefore e _ will be -+0.05. With theae restraints on 0 , 1 + r, and e, the max — o maximum value of £ w ill be limited to 0.0006 degrees or approximately 2 arc seconds. This miaallnement is noted here and w ill be considered 29 to be negligibly snail for the purpose of this study. Therefore the dynamical model and equations of motion for the symmetrical body are considered to be appropriate for the assumed maximum values of e, 8o , and l i l y However, for completeness, application of this small value assumption for the product e(l+r) to equations 2. 10, 2 . 11, and 2. 1 2, and taking note of the heavy dependence of $ and ^ on time w ill result In 9-0 - tane (l+r) cos 2p t . 0 2 o I 4, - + pt (2.18) Applying this same small value approximation to p and 0 gives -h cos© o y " (l+ r)I where 3 then h (I - I 3) p - cos 0o (2.19) “ 3 and (2. 20) For the magnitude of the angular momentum vector, h, constant we have (2.7) Then the two rate terms may be expressed as 30 O ■ —ih tt3 £_ (2. 22) I cos 6 For the case where the lightly time dependent terms of equation 2.18 are negligible or If e » 0 (symmetrical case), then 6 , h, y, and n reduce to the form 0 - 0. h - COB0, > (2 .2 3 ) . (1 - I 3) ♦ - — J » 3 I . “ 3 / I COS0o Effects of a Residual Control Torque on the A s s u m e d Dynamical Model As stated earlier In this chapter, the control system jet thrusters, used to achieve a desired orientation of the angular momentum vector, are assumed to be completely shut off during the stellar observation periods. Here we will assume that the control valves are not perfect and a small amount of cold gas is released during the observation periods, thus producing a small torqulng force. ■ It Is further assumed that a small mlsallnement of the jet thruster orifice exists relative to the principal spin axis. That Is, the thruster Is not normal to the principal spin axis. This mlsallnement can produce a control moment about the principal spin axis which has the effect of producing a torque vector directed along this axis. The effect of such a torque on the dynamical equations of the assumed 31 model has been investigated, and the development of a modified Bet of equations for small cone angle and small torque Is given In Appendix A. The principal results are summarized here. e 9 0 • « 0 • h L ( 2 . 2 4 ) ♦ h /I • e 0 • I - I. h cos 8 (- ♦ II Using equation 2.24 and the definition for oj^ and the definition of h fo r L • 0 , <<>3 ■ ♦ + ♦cose ( 2 . 2 5 ) CO80 (2.7) gives the following 32 where the subscript o denotes the parameter's value for the zero torque case. These initial or zero torque parameters are u _I3“3o "o COB0 0 m (2.27) ♦o " o ' 1 1 M l-l • >1 _*o Lhoco,8o ( I I 3 >J From equation 2.26, is seen to be a linear function of time and the rate of change of this parameter may be expressed as w3 L — - 1 + r - t (2.28) "3o . ho The effect of this small constant torque on the estimation and attitude determination procedure of this dissertation w ill be discussed in a later chapter on flight resultB. However, it is clear that the effect can be Influenced by specifications on the allowable value of L and a choice of t for the stellar observation period. CHAPTER 3 DEVELOPMENT OF A PARAMETER ESTIMATION PROCEDURE In the development of the spacecraft attitude geometry* equation 2.2 was derived to show the relationship between the celestial coordi nates of a specific Identified star and the star's coordinates as defined by the star's elevation angle* n» In either the vertical or slanted slit reference frame. This equation In matrix form is m • m cos 6 cosa COBH cos 6 sina - (*>Om> An observed star w ill produce two equations such as equation 3.1* one each for the slanted and vertical slit planes. The dynamical model considered In this chapter Is that of the torque-free symmetric spacecraft. If this assumed model were the exact dynamical model and If all the parameters of this model were known* determination of spacecraft attitude would be a simple quanti tative task. However* as discussed earlier In Chapter 2 * the true spacecraft model w ill probably contain at least small asymmetries and small residual control torques. The approach taken In this chapter Is to develop an estimation procedure based on the assumed aymnetrical and torque-free spacecraft under the assumption that effects of the small asymmetries and residual torques w ill be negligible. The question of the true effects of these Ignored dynamics w ill be ad dressed through examination of simulation and flight results In later 33 34 chapters. It should be noted that the parameters *, 0 , 0, and e ^ «** assumed to be unknown or poorly known system constants w hile y and 8 are assumed to be well known design constants. The parameters $ and have been defined earlier as ♦ ■ i t + and a ^ ■ ^»t + Hence* the incompletely defined parameters of equation 3.1 are 4* 0* * * 40 » 6 * 'K 4*0, and e ^ In addition to the uncertainties associ ated with these nine undefined parameters* it should also be noted that additional uncertainties are encountered in the measurements of the times at which stars are observed in the optical system reference frame. Thus it is assumed that the uncertainties regarding the assumed model can be defined by a definite set of unknown parameters with system observation uncertainties escribed to noisy time measure ments. It has been demonstrated that (e.g., see Giese and McGhee [ref. 18])* when the uncertainties of a system can be reduced to a finite number of parameters* the classical statistical methods of regression analysis may be employed for the purpose of system identi fication. The formulation of nonlinear dynamic system identification has also been discussed by Foudriat [ref. 19]. In addition Giese and McGhee [ref. 18] have shown that least squares regression* maximum likelihood and Bayesian estimation can be treated from a unified point i of view in nonlinear system identification. I | The estimation procedure developed in this chapter is based on an adaptation of the Gauss-Newton regression method [refs. 18* 20* and 21]. 35 Problems related to the convergence properties of the Gauss-Newton method w ill be discussed later In this chapter. The Gauss-Newton regression method Is also called a linearization technique because the nonlinear prediction function Is replaced by the first two terms of a Taylor series expansion [ref. 20]. In order to derive an explicit formulation, the error cost function considered in this study will be restricted to. a quadratic function so that the nonlinear estimator will be limited to the form of a least squares estimator. Both least squares and weighted least squares estimation algorithms using batch processing techniques were investigated in this dissertation. The use of various sequential smoothing techniques as the estimation algorithm were also investigated. A comparison of three types of sequential least squares filtering methods Is given In Appendix E, along with a comparison of these methods with a sequential Kalman filte r approach. Equation 3.1 may be rewritten as r - JDEHR (3.2) with D - (c2) _1 (e^ " 1 E - W 1 (0)_1 C*)"1 H - ( 0 ) ”1 C*)-1 J - CP)"1 (y)"1 where the matrix elements of the arguments of J, D, E, and H are as in Chapter 2. The in4ivldual elements of the matrices J, D, E, and B 36 are cosy siny 0 -slny cos0 cosy cos0 sinB | (3.3) slny sinB -cosy sinB c o b B j cos ^2 8*n ei sin c2 ~COB ei a*n c2 c o b sin | (3.4) s in e -s in cos e 2 cos cos cos^ cos$-cosB sln$ sinijj cost)* sin^+cosB cos$ sin^ sinty sin 6 E ■ |-sintff cob$-cob0 sln$ cos^i -sini/» sin$+cos0 cos$ cosi^ cos(J/ sine slnd sin$ -sinB cos$ cosB (3.5) sin* H - -cose sin# cosO cos* sinG . (3.6) sin© sin* -sin© cos* COS© The dependence of r on the undetermined parameters may be expressed as r (X) - J D E H R (3.7) where X is defined as the column vector of undertermlned parameters X - <8, *0 , *, *0 . +, *, 0 e1 , c2)T . <3.8) The dependence of t on 7 and 3 is implied in equation 3.7, but is nor. Bhown explicitly, since they are assumed to be well known. The usual procedure followed in solving for estimates of the para meters of equation 3.6 while using the Gauss-Newton regression method 37 Is to express the vector X in terms of an estimate and a correction vector AX [ref. 20]. In this procedure X - X + AX . (3.9) An approximate equation, containing a AX relationship! may be developed by expanding the observation function _r about the estimate % through use of a Taylor series expansion. Thus r (X) « r(X - AX) + F(X - AX) AX . (3,10) The matrix F is the matrix of first partial derivatives of the elements of £(X) with respect to each element of X or Fira " 3X (3.11) m where 1 - 1, 2, 3, m ■ 1, 2, 3, ...| 9 • In all cases, F&m Is evaluated at (X - AX) where AX is some small per turbation of X from Its true value. In the approximation used here, all terms higher than the first variational terms are assumed to be negligible* The vector (X - AX) represents an estimate of X, and AX represents the error in the estimate. This estimate of X will be denoted by X hereafter. Assuming the jth estimate of X> or equation 3.10 may be written for a specific star sighting, say the 1th star, as (X)»jr^ + F^ (5T^) AX*J) . (3.12) Assuming that It stars have been sighted, equation 3.12 may be written as a set of k vector equations 38 r(X)k * rg(j>)k + F(3f(J))k AX(J) . ( 3 . 13) The dimensions of the vectors and matrices of equation 3.13 are r(X )k *► 3k by 1 £(I^)k 3k by 1 F & ^ ) k 3k by m +mby 1 If the error In the approximation of the set of k equations represented by equation 3.13 is defined as then Q(J>k - £(S*J))k - r(X)k + F (2*J)>k AX(J) • (3.14) The quadratic form of the error is QkJ>T QkJ> " ^ ® (J>)k “ r(^ k ] T [-(^ J>)k " *® k] + A- CJ> + AX(J)T F ^* )* j^r(I(J))k - r(X)kJ XT + AX^* Fk AX(J) . (3.15) Using the usual method of a least squares solution, the quadratic form of the error Is minimized by setting its partial derivative with respect to each member of AX equal to zero for a ll m «* (rowJt, F(K(J)y [i® (J))k - £ X + (row SL F(X^^)k F(X^^)k AX^* where % - 1 , 2 m There are nine such equations as equation 3.16, one for each of the undetermined parameters. If these nine equations are written In vector form, the error vector AX^ may be determined as AX(J) - - ^F(?(j))k F(^J))kj 1 F(H(j))k [r(X(J))k - r(X>k] , (3.17) An updated estimate of X may new be determined from equation 3.9 as x«+1> - + tfU> where ? The development so far has assumed knowledge of the elevation angle of the 1th star sighting In either the vertical or slanted slit reference frames, and haa assumed no measurement errors In the observed sighting times. It must be noted that the angle n was variable and not directly measurable. An alternate solution was sought to avoid the necessity of measuring or estimating this parameter. In fact, elim ination of the parameter n can cause the problem to be reduced to an Initial value problem, that is, a problem in which estimates of all parameters may be referred to the Initial time of a time interval spanning k star sightings. Examination of equation 3.1 shows that there are two Independent equations for each star transit time. Also each observed star results In a pair of transit times and a pair of Inde pendent equations for the slanted or vertical slit plane transits. Now If the parameters of equation 3.1 were known, the equations for the slanted slit plane transit could be predicted based on a solution of the equations resulting from the vertical slit plane transit time. This means that knowledge of n Implies knowledge of n and this further V 8 Implies that only two Independent equations exist for each pair of star transit times. The problem solution can therefore be simplified by making use of the constraint of equation 3.1 for each of the two transit times resulting from a single star. This constraint is represented by the second'element of the vector r where (2. 1) Applying this constraint to equation 3.2 results in 0 - J 2DEHR (3.18) where refers to the second row of the matrix J. Now equation 3.18 implies exact knowledge of both X and the sighting time of the 1th star, whereas we w ill have at best some estimate of X and only an observed sighting time which is certainly not errorless. The following develop ment w ill allow for errors In time, as well as errors in the knowledge of X. Let r ^ be denoted by the scalar f^(X , t^) for the ith star sighting for either the vertical or slanted slit reference frame. Now let the observed time for the ith star sighting be denoted by t^ . I 41 Then for small errors In the estimate of X and the observed sighting time the functional f^ (X , t^) may be expressed by a Taylor series as f(i)(X, t4) « f^(jjT^, t^) + Vf^AX^* + Ati (3,19) . (1) where AX *" is as defined earlier, 7 is the ninth dimensional gradient operator and At^ is defined by t. ■ t . + At. . (3,20) i mi i It should be noted that the Taylor serieB expansion of equation 3.19 is again truncated to include only the first variational terms. Hence, the approximation shown in equation 3,20. Note also that In a conven tional sense the error in the measured time Is related to the quantity At^ as Et± - - At^ . (3.21) The mean square value of the quantity At^ will now be treated as being analogous to the mean square value of the error in the observation, or et^. The object of this treatment will be to determine the best estimate of X resulting from a minimization of the mean square value of the error in the observations. For each of the k equations, like equation 3.19, divide by the expression Bf1 8t H(J). ‘nt let each term f*(X, tj) take on its value of zero, and solve for At^. Then each star sighting or observation equation becomes 42 + f Att « 1L af- at at i * Zrrml ( 3 . 2 2 ) Let qt » - Att ( 3 . 2 3 ) where q4 - e tA ( 3 . 2 4 ) and k ■ Ki a f a t jjKj) L ~ » C,mi ( 3 . 2 5 ) For k star sightings let Qk ■ (qjL* q2* •** qk> ( 3 . 2 6 ) or \ - r [ 4 + WK * * (3>] ( 3 . 2 7 ) where 4 - [i1, **, ... £k]T ( 3 . 2 8 ) w ith f1 - f1®1. t .) ( 3 . 2 9 ) V£ Vf (Vf)k - ( 3 . 3 0 ) Vf 43 and 0 .... 0 kg ...» K - • » • # • (3.31) • • • • • • The quadratic form of the observation errors Is l&fc - [ 4 + <7f>k a* Now, as earlier, let the partial of with reapect to each element of AX^ be set equal to zero q X - 0 ■ M (Vf)jKTK4 > (3.33) + (rowi (7f)^KTK(7f)k) AX(j) where 1 ■ 1, 2, 3, ... m Again, there are m such equations as equation 3.33. The Bet of m equations may be written in the form 44 «■ m by 1 For convenience le t K K be denoted as W where W (3.35) The error correction vector now becomes (3.36) It should be remeobered that the matrices of equation 3.35 are all evaluated for X“ ^ and t Q or measured time. As earlier, an updated estimate of X may be determined from the previous estimate and as 3f(J+l) . ^f(J) + AX(j) (3.37) For the Gauss-Newtori regression method the iteration solution process starts by selecting some Initial guess X and then evaluating 45 the corresponding AX'*" from which the new estimate is obtained. ( 1) These steps are iterated until a value AX J is obtained which satisfies a smallness criterion [&ef. 20]. Thus the updating of the estimate of X is continued until each element of AX^ _< M where M is a predeter mined convergence value based on the required accuracy and convergence assurance for the estimates of X. The unfortunate fact is that this iterative solution technique may not converge. It is a common occur rence that convergence is not found in practical problems [ref. 22]. One reason for this lack of convergence Is that the initial estimate « f l £ X places the domain of the expansion of the function f (X, t^) beyond the region in which equation 3.19 is an adequate representation. If this is suspected, a different value of X »1 should be selected and the iteration steps repeated. An alternative is to reformulate the esti mator using hlghest-order terms in the Taylor series expansion of the observation equation [ref. 20]. In practice it has also been found helpful to correct estimates such as X ^ by only a fractional part of AX^\ Without this limitation the estimation extrapolation may be beyond the region where f*(X, t^) can be adequately represented by equation 3.19, and would cause divergence of the estimates [ref. 22]. Various methods have, therefore, been used [refs. 21 and 23] to determine an appropriate limitation on the alse of the correction vector, once the correction direction has been determined. Even so, failure to converge is not uncommon and a number of convergence proofs and convergence techniques, valid under various assumptions about the properties of f^CX, tj) have been derived for the Taylor aeries method used in this study [refs. 18, 24 and 25]. Questions concerning the 46 required accuracy of X, values of the convergence factor used In this study, and procedures concerning alternate selections of or data correction upon failure to converge w ill be discussed In more detail In Chapters 6 and 7. The treatment so far has not Included a discussion of weighted least squares smoothing since we have always treated the quadratrlc form of the errors In the estimate as the cost function to be minimized. The weighted least squares approach was selected for this study because of its properties as a minimum variance estimator [ref. 26] and because the computer programming complexity Is much the same for the unweighted and weighted cases. If we choose for the cost function the expression T -1 Q^R Qk where R is the covariance matrix describing the errors in the observations, then the error correction vector of equation 3.36 can be shown to be AX(j) - - [(Vf)J; KT R"1 K(Vf)kJ -1 (7f)J KT R"1 It^ • (3.38) The matrix KT R * K(7f)J which is the covariance matrix describing the errors in the estimate, will be designated as else where in this paper. The previous discussion Indicated possible sources of time measure- raent errors. One source arises when a point source 1 b passed through an optical system with distortion and through a finite sized slit as used by the star mapping telescope. With extensive calibration and tests It would be possible to define a correction for this error but In most cases the optical system la more easily constructed to make this error negligible. For the electrical noise sources, It suffices to state 47 here that photon, background, and detector noise all contribute to the time measurement error* In many systems these error sources predomi nate and can be assumed to be related to star signal strength. However, no attempt was made in this study to develop a generalized noise model which Is a function of star magnitude. The third source of noise may clock resolution and time counting errors such as dropouts which occur when digital timing systems are involved. While considerable effort has been given to investigation of digital timing errors, no general work is applicable because each timing system usually has distinct resolutions and accuracies. In addition, clock or timing errors may be considered to be independent of the predominant errors related to signal strength and background. A more comprehensive discussion of system noiBes, particularly as related to star signal detection, is given in Appendix B. In the remainder of this chapter the observation time errors w ill be assumed to be uncorrelated and thus the measurement covariance matrix, R, may be expressed as 2 a t* 0 Ot, R (3.39) ot. In this study it is assumed that all observation errors have equal variance. Under this assumption the matrix R becomes 48 R - a2 t (3.40) where I is the k x It identity matrix. For this special case of all measurements having the same uncertainty and variance, the solution for AX shown in equation 3.38 reduces to the form of equation 3.36 AX(3) - - [ Before discussing means of obtaining initial estimates for this problem, it is important that the question of observability or singularities in the F matrix be examined. The problem of near or numerical singularities of the matrix P and the general question of observability for the as sumed dynamical model and observation equation w ill be discussed in the next chapter. CHAPTER A SINGULARITIES IN THE ESTIMATOR COVARIANCE MATRIX, ESTIMATOR MODEL AND INITIALi ESTIMATES OF THE UNDETERMINED PARAMETERS Observability The question of observability in this problem reduces to the question of conditions for singularities in the P matrix of equation 3.36. The condition for non-singularity can be expressed as [ref. 27] d et P - det l(V f)J KTK(Vf)k ] > 0 . (4.1) As stated earlier» k (the number of observations) must be greater than m (the number of undetermined parameters) for the case of noisy obser vations. It can be noted that, for no noise, k may be exactly equal to m. For the case of no noise (or no errors in time), the condition for non-singularity reduces further to det P - det t(Vf)J (Vf)kJ > 0 . This condition for non-singularity is equivalent to treating the P matrix as the Gram matrix and applying the test of linear Independence to the vectors of (Vf)k [ref. 28] 49 50 af1 af1 af1 3xx * 8X2 ’ ' 3xm (Vf), (A. 2) 3fk afk 3x. » 1 ax 1 m For convenience, let the column vectors of equation (4.2) be denoted by jt 1 , where 1 “ i a f 1 ZJJL 1i 3 x i 2, * 2 m , 1 “ 1, 2, . . . , m (4.3) 3 f k 14 k 3* t Determination of the conditions for linear Independence of the set of m vectors, z1, will be established by determination of the conditions for the set of vectors to be linearly dependent. Now, linear dependence can be examined by determining If a set of constants, (not all zero), can be found to satisfy the equation [ref. 29] 1M c. z1 ■ 0 . (4.4) 1-1 1 “ Before proceeding, it should be emphasized that we desire to determine the conditions under which star sighting data can be used to obtain or estimate those parameters which describe vehicle motion or attitude. For convenience, the parameters and e 2 will not be considered In this portion of the analysis. 51 An element of Che vector z_ may be expressed s b i M rB m 2 3 XI - (4.5) where X - (note that R Is not necessarily the same vector for each element of £*) £ l - • » » ! ! “ £3 ’ J2 ff « ■ J2 If" t / (4.7) * 7 • Expansion of the equations (4.7) gives B31 s in E32 s in \Ji E33 sin . i . E31 cos 4> E32 c o s E33 c o s 4* H R (4.8) s in $ -E33 c o s $ - sin 6 mE33 " E21 E22 E23 «2 - j . H R (4.9) - * u - E12 ”E13 0 0 0 52 3 2 « ■ t i (4 . 10) 0 -*12 E11 0 H R 4 * - J2 -K22 E21 (4.11) 0 -*32 E31 «i £ 5 - t i * (4.12) 0 ”*12 H11 6 z ■ 0 (4.13) •V -*22 H21 0 *“*32 H31 * 0 0 r 7 - JjE H H31 H32 33 (4.14) -H "H21 H22 23 Since the vehicle is assumed to be untorqued, the angular momentum reference frame is fixed in inertial apace, and therefore can be con- aidered an inertial reference frame. For convenience, the values of a and 6 can be selected to simplify the matrices of equations 4.8 through 4.14. In so doing the star vectors R are still perfectly general. For simplicity let $ be zero degrees and 0 be 90 degrees. Then the m atrix H is 1 0 0 H 0 0 1 (4.15) 0-10 53 The vectors of equations 4.8 through 4.14 become ■ « E slnty slni|) E1^2 Bin4> z1 - J, E ^ cos<{; -E!33 cosi|> E32 cos^i (4.16) E^^ sln E E21 "E23 22 z2 - J, -E (4.17) “El l E13 12 0 0 0 i (4.18) 0 - E12 E11 z* - J, 0 “E22 E21 (4.19) ~E32 0 E31 z< (4.20) 0 ' E13 E11 JE® - J, 0 E23 E21 (4.21) 0 E33 E31 0 "E12 ”E13 z 7 « J , 0 *~E22 ”E23 (4.22) 0 ”E32 "E33 Now In order to show linear dependence, it Is necessary that 7 i C ci - * ° 1-1 54 where not all are equal to zero. For exanple, let c^ - ■ c3 - c5 - 0 . (4.23) Then for any a tar we can write for the n ^1 element of all vectors c .z 4 + c ,z 4 + c-z^ ■ 0 . (4.24) 4 n 6 n 7 n Collecting terms of equation 4.24 gives *J21E13 + J 22E23 + J 23K33*(c6Rl “ C7R3* + (J 21E11 + J 22E21 + J 23E31) ( c4R3 + C6R2) - (J21E12 + J 22E22 + *^23E32^(c4R! + C7R2> " 0 * (4 *25) For convenience let c^ be unity. Then equation 4.25 can be satisfied for the conditions h . C6“ _R2 (4.26) R1 C7 " " R* Mote that for a single star the conditions for linear dependence are p resen t. However, i t should be remembered th a t more than one s ta r will be observed. There do not appear to be any additional linearly dependent terms. Although £ 3 and z_ 5 are related to z 2 and z_ 4 respec tively by the factor t, it must be pointed out that this t will be different for each n or star sighting. The inclusion of Cjjs* in this analysis merely places sore stringent conditions on the star coordinates by requiring that 55 *J 21E21 " J 22E11^ R1 “ + (J2l E22 " J 22E12) R3 " 0 * (4.27) Since Che E matrix la time dependent, equation 4.27 would require that R be time dependent in a peculiar manner to continuously satisfy the condition of this equation. Hence, it would appear that the conditions of equation 4.26 are the only ones capable of satisfying the linear dependency criteria for unobservability. Near Singularities for Small Cone Angles A single row of the (Vf)^ matrix, resulting from a specific star sighting, can be written as (Vf) _J §___ §___ 5___ 3__i a 8 a_ (4.28) axj^ 3x2f 3x3 » 3x4 » 3x5 » 3x6 * 3 x?* 3Xg* 3xg J2U - where the partial derivative operators are defined as mrn mm 3 " 3 ' 3x^ 36 3 3 3x 2 * 0 3 3 3Xg 34> 3 3 3xa 3^o 3 3 (4.29) 3xs 3$ 3 3 3x 6 36 3 3 3X7 36 3 3 3x 8 3E1 3 3 ■3x9 ■5^2 m «■ s* m 56 Equation 4.28 la a general expresalon for any star transit time where the m atrix E la the only m atrix dependent upon 9. The problem a ris in g from a zero or small 6 may be Illustrated by examination of this m atrix E. The m atrix E may be w ritten as B(4,9,4) ■ A (4 + 4) + sln 0 + (1 - cose) CU,*) (4,30) where ‘ COB(4+4) s ln ( 4+4) 0 A - - s in (4+4) cos (4+4) 0 0 0 1 . 0 0 sin4' B - 0 0 cos 4 . sln+ -C0S$ 0 . Bin4 sln4 -cos 4 eln4 0 sin 4 coat|» -cos 4 cos 4 0 -1 It Is clear from equation 4,30 that for zero 6 E(4,6,4) - A(4 + 4) (4.31) and 3E 3E 36 34, (4.32) —r3E 3B 3E 34 3$ 34 The conditions of equation 4.32 Indicate that two pairs of columns of the (Vf)^ matrix will be equal and the P matrix w ill be singular. It is interesting to note that a similar difficulty exists for zero 0 . For this case the Euler angles ♦ and are indistinguishable and therefore two columns of the (Vf)^ matrix will be equal, since df 3f .. o Now for small 6 , for which sin 0 and (1 - cos 6 ) are small, 3E 3* o (4.34) 3E 3E « t - ■=£ o Then there are two p airs of columns of (Vf)^ which are n early equal. The situation of a non-zero but very small 0 results in an approxima- i tlon in place of equation 4.33 and causes two columns of (Vf)^ t0 be nearly equal. In this study the problem of small 0 was ignored since it was assumed that either the launch could be controlled to achieve a reasonable value of 0 or some other arbitrary reference frame could be selected. The latter approach would, of course, require that the cataloged stars' coordinates be transformed appropriately. However, the problem due to a small cone angle 6 i s indeed re a l sin ce good design of spin stabilization systems leads to cone angles of six degrees or less. In thlB study the cone angle will be assumed to be no larger than 0.5 degree. The physical reason for the problem introduced here for either zero or very small 9 is that the angles and are poorly defined individually and only their sum is well defined. The approach taken 58 here is to find a new vector AY such that AY will contain sums of the • * parameters A$qI A^q, A$ and A4*. A modified least squares problem will then be solved for Alf. This solution will be transformed to AX from which w ill result the sought-for estimate of X. First, examine an expansion of equation 3.19 for a specific star n0 « A f a +. 3fi An AOj. + —— 3^ A^ia r + a. t► .^« Ad* + . , AA j. + .t . 3^t .~* A AA1 30 3iJjo To i 3i|)q T 39q To i 34io T , it1 . sf1 . at1 , . i t 1 , . at1 + i T 4* + 30" 40 + 9 ^ 4cl + 3^ 4cZ + 3F 4ti + "• • (4,35) Adding th e mixed terms 3 f l a a 4 , 3fi aI 3fl AA - 3fl aI n n W**oo + hW** To ~W To ^o ~ hW**r0 0 (4,36) to equation A.35 results in 0 V 1 + I r - 46 + !*; (4*o+ 1V + ci <4* + 44> , /af1 3flN\... . / af1 af1 \ Aj. of1 A* V *o " 3* J ° 4 V 4o " 3*oj 84 . Sf1 . af1 . . it1 . . it1 ,,, + 80“ 40 + 3^ 4cl + 4c2 + aT 4tl + • •' ' (4,37) The fa c to r (it1 af£\ V *o ‘ **o) of equation 4.37 can be expressed as 59 /wi. niV , „ /isi _ »i\ „.(« \ 5*o 8V 1 V*o 9* o / - J,D[oln6(B^ - bJ ) + (1 - C086)(cJ - cj )) II Ru> ®o M ta To - 2sin | J2D[C08 |(bJ - bJ ) + sin |(cj - cj )]HR(i) (4.38) where the eubacripts on the B and C matrices denote partial differen tiation with respect to the subscript variable. With the use of equations 4.37 and 4.38, a new correction vector, AY, may be defined as A6 A4To + Atfi r o a * + a {* A« AG . (4.39) 2 9ln f 2 sin y A$ Ati At. The modified (Vf) is then 60 ” J2D(coae B + ein0 C) HR J2D(A. + sine B + (1 - cose) C ) HR vo vo tJ2D(A. + sine B + (1 - cose) C ) HR J 2 0 E H4 R J 2 D E H0 R (A.AO) J D co.(f + sin fee - C ))SR ' To o To r o ' t J 2D cob (|< B - B ) + Bln |(C - C )) HR ' o ro Yo ro' J , D E H R 2 El J . D E H R 2 where the subscripts again denote partial differentiation with respect to the particular subscript. The equation A.39 indicates that estimates • • of the individual components A$Q, A ” e " yl a* o Ay2 - Ay^/2 sin J- a* Ay3 - Ay7/2 sin |- A6 Ay^/2 sin y To AX - A 4. ■ Ay7/2 sin |- (4.41) Aft Ay4 AO 6y5 1 Ae^ Aya ^ e2_ Ay9 For very small 6, soy, leas than .01 degree, no attempt was made to ■ estimate A4Q and 4$. For this case, "V Ay2 4y3 0 AX - 0 (4.42) Ay4 4y5 AyB Ay9 Initial Estimates Before any solution is possible for the nine undetermined para meters of X, in itial estimates are necessary. Of the undetermined parameters, e^, e2 and 6 were assumed to be eero; +0 and ^ were assumed to be completely unknown; 6 and 6 were assumed to be known to 62 w ithin + 10* from available launch data; was assumed to be known with an accuracy of one or two per cent from observation of repeated sightings of the brightest star of all detected stars; and $ and 41 were assumed to be known to accuracies dictated by the pre-flight measurements of 1^ and 1^, and the limitations of the assumed model. With these limitations on the initial estimate, a reduced problem of three undetermined parameters is posed* The undetermined parameters of this reduced problem are + i^Q), 4 and 0 . Even with this reduced problem we are s till faced with a problem of determining some initial estimate of ($o + The initial estimate for the three parameter problem was determined by uBing equation 3.1 with the assumption chat yv was zero and setting the vertical slit transit time for the first identified star to zero. In addition, the elevation angle ny for this star was assumed to be zero. Equation 3.1 then becomes cosS. cosa, cos 6 1 sin a. (♦) (e) <*o) <*o) ^ 0 (4.43) s in 6 "cost -cos© s in t sin© sint* cos($o+i|io)- - s in t cos© cost -sin© co st sin($o+i|jo) . 0 sin© cos© 0 iere the subscript on a and 6 denotes the right ascension and declina tion values of the first identified star. The matrix equation 4.43 may be solved for (6 o + tb o ) from 63 -cob6^ cob0 s ln ( t - c t j ) + sin0 sinfi, . (4.44) tan (*Q + *o) - cos6^ cos(6 - a^) In this three parameter problem, the AY vector Is ' o" Ay2 0 Ay* AY - Ays (4.45) 0 0 0 0 where the elements Ay 2, Ay^, and Ay^ are determined by the previously discussed least squares procedure. The solution for the three parameter problem produces In itial estimates of ($Q + ^Q), which are Improve ments of the estimate obtained by the use of equation 4.44, and Im provements In estimates of 4 and 0 obtained from launch data. It should be noted that this procedure can give Initial estimates of but not their Independent values. The nine parameter estima- ♦ tlon problem was Initiated using the results for $ and 6 obtained from the three parameter problem. The Initial estimate for £ + was set equal to the Initial estimates of defined here as Initial estimates for 6, e^, and e2 were set at zero degrees. The initial estimate for 40 was set at Thia procedure, although arbitrary, waa not unreaaonable alnce we have already shown that the nine parameter problem Is heavily dependent on 40 + 4*0 and lightly dependent on their Individual values for small cone angles. The effects of equation 4.46 on convergence will be considered later In a discussion of flight results. CHAPTER 5 DEVELOPMENT OF A MODEL FOR GENERATING SIMULATED STAR TRANSIT TIMES Before applying the discussed attitude determination technique to the processing of flight data, the accuracy of the parameter estima tion procedure must be evaluated In order to establish expected attitude determination accuracies for a particular set of dynamics. In addition, some criterion must be developed in order to assess the adequacy of any given estimate of X as determined from flight data. These questions, particularly the question of expected accuracy of spacecraft attitude determination, were examined in this dissertation through the use of simulated star transit time data generated by an assumed model of spacecraft dynamics. This chapter discusses the development of this model and digital computer program used In the simulation studies. Discussion of these simulation studies and results are presented in the following chapter. The dynamical equations describing the motion of the spinning vehicle and the geometrical relationship of the star mapper reticle to the celestial reference frame, as discussed In Chapter 2 and Appendix A, were used as the basis of a model for the generation of simulated flight data. As described earlier, the transit times of stars were constrained to be those which could be transited by a slit plane, 65 66 parallel to the principal spin axis, and bounded by vectors located at angles of + 3° relative to the star mapper optical axis. This 6° by 6* fleld-of-vlew constraint plus a restriction that transited stars must be +3.5 visual magnitude or brighter for detection were helpful In keeping the number of candidate stars down to a reasonable number. This visual magnitude constraint is discussed In Appendix B, where It 1b shown that this limit ensures that at least three stars will be "seen" during each revoluation of the spacecraft. The orientation of the body with respect to the celestial refer ence frame Is described by the ten reference frames of Figure 2. The basic equations relating the coordinates of a given star In the right ascension (a) and declination (5) coordinates of the celestial reference frame to the elevation angle n In the vertical slit plane are (see equations 3.1 and 3.2) " cosn “ 'cosa cos6’ 0 - J D E H sina cos6 (5.1) _ sInn . sln6 where J - J ( y ,B) D - D(e1,c 2) E - E ( M .’J') and H - H(4>,0) The angles of the arguments of J, D, E and H are defined In Chapter 2. However, it should be noted here that a simple set of relationships exists between the angles $ and 6 and the right ascension and declination of the angular momentum vector, k^. From Table 1 and \ / 67 equation 2.3, the coordinates of expressed In the (ij,j^,£^) reference frame are sin# BinO -cos# sinO ( 5 . 2 ) k’3 COS0 and cosci cos 6 sln a cos6 ( 5 . 3 ) sind where a and 6 of equation 5.3 represent the right ascension and declina- tlon of k^. From these two equations it is easily seen that G - 90 ° - 6 and • - a + 90° The objective of the model development of this chapter is to provide a means of computing the time of transit and elevation angle n of all stars as they cross the defined but unconstrained vertical and slanted slit planes. Then, through use of the field-of-view constraint of 6° by 6°, all stars not transiting this limited field of view may be eliminated. It should be remembered, of course, that this field of view is scanned over a 360° sector of the celestial sphere for each spacecraft revolution. We will first compute the time for which known stars transit a slit plane for y ■ 0 ■ 0°. The expression of equation 5.1 is satisfied the instant a star of known a and 6 transits this vertical slit plane (i.e., y ■ 0 ■ 0). For simplification, the 68 following terns of the matrix product JDEH are defined: P ■ sin® cos$ + coa0 sin® c o b ® Q ■ sin® sinij) - cosG cos® cos® (5.4) R ■ cos® sin0 Using the definitions of P, Q, and R, the second element of the m atrix equation 5.1 may be written for the assumed vertical slit plane conditions of v - 0 ■ 0 as V 0 “ a^ cos6 coaa + a 2 'cos6 sina + sin® (5.5) where a^ ■ -P cos® + Q cos© sin# + R sin© sin® a^ - -P sin® - Q cos© cos® - R sin© cos® (5.6) and a^ ■ -Q sin© + R cos© From Chapter 2 we are able to write 0 - 0 - j tan0 Q (1 + r) cos o ®t ■ ®Q + yt ® - *o + (2.18) ®t - ®Q + ♦ " *o + e (I - I3) cos0o where y * ® m ( 2. 21) I u3 cos0 and n - ® • - ■=* *3 w3 I cos0 (2 .22) The principal equation 5.5 may be written as 0 - fla^eOO, ®(t), ®(t), ®, ©), a2( ), a3( )] (5.7) i 69 to emphasize the functional dependence on time. Since time appears as a linear factor In COB6 - [1 - (1 - CO80)] , (5.8) the terms P, Q, and R may be written as P ■ sin (iJj + $) - (1 - cose) sln^ cosij/ Q ■ -cos 0|> + ♦) + (1 - cos9) cos$ C08lj« (5.9) R * costy sin6 . Since (1 - cosO) and sln0 are expected to be small (of the order of 0.00004 and 0.0087 respectively for an expected 0 value of 0.5 degree), we will Isolate the terms involving these quantities. Using equation 5.9, equation 5.5 may be rewritten as 0 ■ a sin(i(i + $) + b cos(i|> + $) + c (5.10) where a ■ -cos6 cos(4 - a) b ■ cos0 cos6 sin($ - a) + sinO sinfi and c ■ cos^ (-(1 - cos6)[a sln$ + b cos$] + ain9[cos6 ainG sln($ - a) + slnd cosG]} . Since the time dependent quantities appearing In c are always multi plied by (1 - cosO) or sln6, it may be assumed to a first approximation that for small 0, c has little dependence on and $ and hence depends little on time. Equation 5.10 may then be solved by a simple replace ment scheme fo r the angles (4> + $) and consequently the time t . S ta rt by setting t ■ (K - l)r , where t Is the spin period time and K 70 represents a spin period number. It should be noted that the times of interest during each spin period are (K - 1)t < t < K t , K - 1, 2, 3, ... . Choosing a time, t, $ and \|t are calculated and used in c only, after which the coefficients a and b are computed. With the coeffi cients a, b, and c computed, equation 5.10 may be solved for ($ + \|>) by solving successively for sin( sln($ + ijO and cos($ + ifi) are -a c - b \/a2 + b2 - c2 sln<* + *-b Z— (5.11) (a2 + b2> -b c + a V a 2 + b2 - c2 and cos ($ + *) - , I — . (5.12) (a + b2) Note that the algebraic signs before the radicals in the expressions for aln(4 + and cos(4> + V1) must necessarily be of opposite sense In order to satisfy the condition that sin2($ + t(<) + cos2($ + \|/) • 1 . Using equations 5.11 and 5.12, the expression for tan ($ + \|i) becomes /. . .v -a c + b V a2 + b2 - c2 , ... tan($ + ^i) - ...... d . (5.13) t / 2 ^ 2 2 -bc-ava -f b - c The so lu tio n fo r ($ + \p) within any spin period is then * + if) - 2tt(K - 1) + tan-1 d (5.14) 71 or $ + ** + ijit + <|»o + ($ + ip) - (4»q + ^0) Then t - (5.16) 2r(K-l) + tan-1d - U Q + tj<0) or • * (5.17) Thus, by fixing the quantities 0, iji^, IhlB scheme will produce a time of transit of a selected star across the vertical slit. Now it becomes necessary to compute the elevation angle n end the transit time across the slanted slit illustrated in Figure 3. The angle n can be obtained by considering the first or third elements of equation 5.1 from which is obtained sinn " cos6 cosa[sin0 sin$ cosfl> + sine cos$ cosO sin* + cosO sinO sin*) + cosfi slna[sin9 sin* sin* - sinG cos + sin6[-sin0 cos* sinG + cose cosO] (5.18) Simplifying equation 3.18 gives sinn “ sin(* - a) cosfi[cos0 sinO -4- sinO cos* cosG] + sinO sin* cos5 cos(4 - a) + slnfi[cos6 cosG - sinG sinO cos*] (5.19) i ! i 72 An equation similar to equations 5.18 or 5.19 may be written for the cosine of n* The basic relationship of the reference frames of interest and the field of view elevation angle n has been described and illustrated in Chapter 2. The solutions for n for all transited stars will fall within the range |90° - T - n| 1 3° * A where T is the angle between the kg axis and the ig axis. The angle T 1b equal to 90° for perpendicularity. The relationship of r and n is illustrated in Figure 5. Having arrived at a transit time for the vertical slit passing through the optical axis and parallel to the kg axle (i.e., a slit defined by Yv ■ Py ■ 0), and having determined the angle ny for the vertical slit, we must now find the star transit time for the slanted slit. The equation to be satisfied the Instant a star falls in the slanted slit Is again, as defined In Chapter 3, J 2 DEHR - 0 (3.18) where the vectors J2 and R and the matrices D, E and H are as defined in Chapter 2. Expanding In a Taylor series about an arbitrary time t, equation 3.18 becomes J2DE(t)WR+ ~ J 2DE(t)HRAt + ... - 0 . (5.20) Since only E contains time-dependent quantities, the partial derivative can be written as jj- J2BB(t)HS - J2D ( | f 6 + | | * + | |i) HR . (5.21) Principal Spin Axis 4 i„ Optical Axis S lit Figure 5. Relationship of the Telescope Mounting Angle, T, to the Elevation Angle, n* u At Direction of 1th, s ta r Slanted s i l t plane Vertical silt plane Figure 6. Geometrical Relationship of the Transit Tlaes of the Vertical and Slanted Slit Planes. 74 Ignoring the second order terms and higher, the correction to any previous estim ate of t becomes - J 2DE(tn)HR Atn " /aECt) . 3E(t ) . DE(t ) (5,22) — HR J 2° y ~ i r - 6 + — * r * + a* and S r il " *n + Atn <5*23> where t Is the previous estimate and tfl+^ Is the corrected estimate. This process should converge since the initial estimate for t is so good. This initial estimate la accurate because the transit of the ith star across the vertical slit has been previously determined and very little precession will have taken place since the two slits are so close together. To obtain the first estimate of the slanted slit transit time, consider the slit projections to be on a plane as illus trated in Figure 6. Given a transit time of a known ith star across vi the vertical slit plane, tQ+1 , an Initial estimate of the delta time required for the star to traverse the field of view is Atl * ^ » Y8 < 0 ‘ <5‘24> A first correction to this initial estimate may be set by * h tan B «i - — • <*•*» Of course, At2 Is zero if n is zero and At2 is a negative quantity If n is a negative elevation angle in the field of view. A good first estimate for the expected time of transit of the indicated 1th star 75 across the slanted slit plane Is then vi * tan " V6 t~"s i - t” , + ------5------o n+1 w_ (5.26) Using the time calculation of equation 5.26 as an initial estimate, successive Iterations of equations 5.22 and 5.23 should yield the correct transit tinelie of the slanted slit plane to within some specified change in AtQ (say, a fraction of a microsecond). It should be noted here that u,, 0 , ar 4 8 nd Y_D Are known and n has been calculated for the vertical slit transit using equation 5.19. In the manner described, given the spacecraft dynamical model parameters and selected stars In terms of their right ascension and declination coordinates, transit times may be determined for both the vertical and slante^l slit planes. Given these transit times, the values of n and n v a may be determined, and all stars having elevation angles n which fall outside the bounds of the field of view would be rejected since they would not have been "seen" by the star mapper. Further reduction of candidate Btars Is possible through rejection of sta rs dimmer than sqme sp e cified threshold v isu al magnitude. Using the mode of the previous discussion, a digital computer program was written for the generation of simulated transit time data, Inputs to the progra m include the following: 1. A star catalog Including stars to a limiting magnitude as determined by in stru lament design. Star information available to the program was the Boss Catalog [re f. 30] number of each s ta r used, i t s visual magnitude, aqd its celestial coordinates (right ascension and declination); and 76 2. Vehicle and tilt configuration parameters: a. length of time the simulation was to cover; b. slit separation angle, angle of the slanted slit, field of ▼lev lim its, and the angle from the principal spin axis to the optical axis; and c. moments of inertia, asymmetry parameter c, spin rate about the angular momentum vector, in itial orientation of the angular momentum vector, and the set (0O»'+O* ^0» Q» c^> e2^’ It should be noted that provisions were built into the transit time simulation program to add normally or uniformly distributed random errors (with zero mean and standard deviation, ofc) to both the transit times and star magnitude data. This latter provision was used to simulate errors in instrument measurement of magnitude and also to- provide a set of data with errors for testing the star identification procedure discussed in Appendix D. The data representing transit times plus* time measurement errors were used in a series of computer studies to determine the effect of these time errors on both the accuracy of the undetermined parameter estimation procedure and attitude determina tion as obtained using the parameter estimates. These studies, which are discussed in the following chapter, make use of a digital computer program developed for the model of the estimation procedure described in Chapters 3 and 4. Acknowledgement must be given to the staff members of the Analysis and Computation Division of NASA Langley Research Center and particularly to Mr. D. £. Hinton of the Flight Instrumentation Divi sion for their support in the development of these computer programs. CHAPTER 6 SIMULATION STUDIES In the simulation studies, the mathematical model describing the equations of motion and physical configuration of the simulated space craft was selected to compare closely with like parameters assumed for a flight vehicle. These assumed spacecraft dynamics and physical des cription, along with assumed launch conditions and time of launch, were applied to the simulation program described in the previous chapter. Simulated flight data consisted of a listing of the star transit times and the right ascension and declination of these stars updated to the assumed launch time. These simulated data with appro priately added time measurement errors were used to estimate the pre viously described parameters of X which in turn were used to determine attitude or pointing direction of selected axes. Errors other than star transit time measurement errors that can affect estimation accuracies are errors in the assumed mathematical model of the system and computational errors. In this chapter the dynamical model, used in the generation of transit time data, Includes a small asymmetry between the moments of Inertia 1^ and while the dynamical model of the estimator is that of a symmetrical body. -The effects of this error in the assumed dynamical model of the estimator i on the accuracy of pointing direction computations are Included in the results of this chapter. The effects of additional errors in the assumed dynamical model of the estimator are included in the discussion of flight results. The effects of computational errors on estimation 77 78 accuracy were investigated for the case of no measurement errors. For the case of no measurement errors (zero transit time errors), computational errors were found to be sufficiently small that the nine parameters UBed to generate simulated transit time data were recovered with nearly zero error using the previously described estimation procedure. The ideal attitude measurement device provides an error-free measurement of the vehicle attitude with one set of data being suffi cient to completely determine the vehicle states for all time. However, instrument noise and resulting uncertainties in vehicle motion require that the attitude determination of the vehicle be treated as a stoch astic process. Since the attitude determination measurement Bystem is a stochastic process, the determination of the vehicle attitude becomes more complex. The initial conditions must now be determined by data obtained from an imperfect measurement device, and applied to equations of motion which may not describe the exact behavior of the vehicle. Consequently, an uncertainty in the determination of the Initial condi tions of the vehicle's state leads to errors in attitude which must be periodically corrected, i.e., the initial conditions must be statisti cally determined, periodically updated, and a new solution begun. This chapter presents the results of a Monte Carlo study of the effects of time measurement errors on the accuracy of attitude determination. In this study, the errors in the pointing direction (in terms of A right ascension and declination) of the ig axis were the basis used in determination procedure. The results of the simulation studies were also used to develop a criterion for assessing the adequacy of any given 79 estimate of X* The star mapper data processing procedure was investigated further to determine the feasibility of predicting or extrapolating in time the pointing directions or orientation of the lg and kg axes based on the estimates X. The two principal mevbers of the Bet of nine estimated parameters which affect the ability to predict attitude r r were $ and $ since errors in the estimates, multiplied by expired time, t (tQ defined to be at the beginning of a data fitting period), are directly reflected into errors in right ascension and declination of A * the ig and kg axes. This fact limited the usable extrapolation period to values of time close to the data fitting period. It was decided therefore to confine attitude determination computations to values of time falling within the data fitting period. A discussion of the extrapolation study, particularly error propagation due to rate errors, is given in Appendix G. Monte Carlo Study; Expected Accuracy of Attitude Determination Procedure Figure 7 shows a flow diagram illustrating the generation of time histories used in the Monte Carlo study. A selected set of nine para meters defining the equations of motion, assumed launch data, a parti cular reticle configuration, and the right ascension and declination of cataloged stars was used to generate simulated star transit time data. The simulated transit times plus expected errors and the right ascension and declination values of the transited stars were processed by the parameter identification program in the same manner as flight data. In addition, the a priori estimates used in this processing were identical to those used for flight data processing. The values of the nine Launch time, vehicle orientation, and reticle configuration Right ascension and declination Nine parameters Star transit defining equations time simulation of motion program Star transit time Time e rro rs . Time e rro r generation Estimates of the program nine parameters Time h isto ry program for _> Ea(t ,1 8 a , 6, and pointing ■* c5(t)i8 direction for selected axes -> cp ( t ) i 8 CD Figure 7. Flow Diagram of Error Time History Generation. o 81 parameters determined by the parameter estimation program along with the known values of 3y, yv» and Ys were used to generate time histories of the pointing direction of the star-mapper optical axis. The variation of error In this pointing direction as a function of several parameters was investigated and is presented here* These para meters Include the length of time covered by a least-squared data fit in terms of fractional parts of a precession cycle, number of stars per spin period used in the data fit, and angular separation of stars within a spin period. Since the Information provided by the star mapper was transit times of observed stars, the basic errors in the system were inaccuracies In the detection of transit times for the vertical and slanted slit planes. The two sources of error which contributed to these transit time errors were the inaccuracies In time detection of a star transit across the slit planes and resolution of system clock time. A discussion of these error sources and their causes Is given in Appendix C. Differ ent sequences of random time errors were added to each selected sample size of the simulated data. Thus, for each sample size of star data, a run through the least-squares program was necessary for each noise sequence to determine the resulting estimates of the original nine para meters specifying vehicle dynamics. Each set of nine parameters.was then fed Into a tlme-hlstory generator containing the vehicle dynamic equations. Beginning at time zero, and for equal intervals of time, a time history of attitude data in celestial coordinates (right ascension * and declination) for the star mapper optical axis ig was calculated for each estim ate of X. Each of these time h is to rie s was computed fo r the length of time covered by the selected sample size of simulated data 82 used in Che least-squares program. This computation provided time- history records of attitude both at the times of star sightings and during the periods of time between star sightings. The attitude data representing times between star Blghtlngs provide a measure of the interpolation capability of the attitude-determination procedure. Each time history associated with a transit time noise sequence was compared with a reference time history generated by use of the original nine parameters used in the simulation program. This comparison re sulted in an ensenble consisting of as many error time histories in right ascension and declination for the ig axis as the number of noise sequences used in the generation of polntlng-direction time histories. As indicated in Figure 7, these error time histories are represented by the set of errors * Ea ^ 18 * c6 ^ 18 * An expression for the total pointing direction error for any axis at any time t may be expressed in terms of the error terms cq and e^. This expression is developed here. Let the coordinates of ig(Xf t) in the (i^, system be expressed as 'c o s 6 cosa lg(X, t) ■ cos6 sina • (6.1) sinfi Let the coordinates for this axis as a function of any estimate St be expressed as 83 COS 6 COB o’ i 8& t ) cosT sino (6 .2) sin6 where o + e (6.3) and ? + (6.4) Note that the components of ig(X,t) and ig(X,t) can also be expressed In the (l1»j^»k^) reference frame In terms of the matrices D, E, and H as ,T “ I^ E K]' (6.5) X and l g & t ) ■ [DjE H]1 (6 . 6) where the subscript 1 on D indicates the first row of this matrix. The correct right ascension and declination angles shown in equation 6.1 and the estimates of these two angles shown in equation 6.2 may be found by -1 A8 * h a£g(X,t) - tan A A (6.7) i 8 * *1 or -1 DjE H a £0(X ,t) - tan (6 . 8) DjE H where the superscript on H Indicates a particular column. The angle d£g(X,t) may be determined by 84 «£8(X ,t) - sin "1 [DjE H3] x (6.9) The expressions for a£g(X,t) and 6£g(X,t) are identical to equations 6.8 and 6.9 except that the matrices D, E, and H are evaluated for jx and t. The expression for the right ascension and declination angles, which define the pointing direction coordinates of any selected axis AAA in the (^ J^ * ^ ) reference frame, may be determined In a like manner A A as for the ig axis. For example, the components of the kg axis in the (±1»Jl.ici) reference frame are (6 . 10) (6 . 11) The values of a£g(X,t) and 6£g(X|t) are determined by use of equations 6*10 and 6.11 with the matrices E and H evaluated for Jt and t. The angular displacement, ep, between the two vectors of equations 6.1 and 6.2 is found from (6 . 12) ig(X,t) * i8(3[,t) - cos Ep cos Cp ■ costi cos? cos(a-a) + sinfi sin? . (6.13) Substitution of the expressions for o' and 6 of equations 6.3 and 6.4 into equation 6.13 gives cos eD ■ cos6 cos(6-e.) cos c + sin6 sln(6-e.) r o a o (6.14) 85 New with the assumption of small e^ and Ep, equation 6.13 may be reduced to 2 2 2 cp » e , + e coafi (cosfi + e sinfi) . (6.15) r o a a Now the declination angle Is restricted In the application of this study to 0° ^ J6 J 60°. Equation 6.15 may be further approximated as 2 2,22, , , , e„r ** e. o + c a cos 6 (6.16) or Ep * ^ + eq^ cos^fi . (6.17) The expression of equation 6.17 Is used throughout the remainder of this dissertation to represent the total error In the pointing direction of specified axes. It Is appropriate at this time to discuss the basis for the selection of the number of sequences used In the error analysis. i Basis for Selection of Ensemble Sample Size The procedure used to establish this basis was to examine the magnitude and convergence character of the ensemble statistics of the A pointing direction error of the ig axis for ensemble sizes of 10, 15, 30 and 54 error time histories resulting from the use of a like number » of time error sequences. The values of the nine parameters of X> the vehicle Inertias and the reticle parameters used In the generation of simulated star transit time data are given In Table 2. 86 TABLE 2 SIMULATION STUDIES INPUT DATA Values of X e - .310758* r 0 - 51.081051* - 316.572701° ** ** 287.844975 deg/sec • * ■ 19.137575 deg/sec ♦ - 76.462935° 0 - 54.126671° el - .064170° e2 - .031017° Reticle Parameters « ■ Y o V e ■ 8 o V - -3.071010° Ys 8 - 43.015970° s Vehicle Inertias I - 129.92 slug - ft2 e ■ +0.05 " 8.27 slug - ft2 87 The value of for these values of Input data becomes 306.982268 degrees/second. A partial listing of the simulation program printout of detected stars and their transit times Is shown In Table 3 and a partial listing of the star catalog data used in this simulation is given in Table 4. TABLE 3 SIMULATION DATA LIST SAMPLE S tar Number Transit Time Visual Boss Catalog (seconds) Magnitude Number A8 .0000990002 1.700 6668 AB .01000566AA 1.700 6668 63 .0656448520 3.200 7969 65 .0695323750 3.190 8208 65 .0758920677 3.190 8208 63 .0763643620 3.200 7969 95 .2003760125 3.120 12407 95 .2060111099 3.120 12407 102 .2212026585 3.260 13157 102 .230880418A 3.260 13157 113 .2655338539 2.440 15145 113 .2823157239 2.440 15145 120 .2903425392 2.540 16268 120 .3007017906 2.540 16268 The first transit time of each transit time pair Is for the vertical slit and the second la for the slanted silt. Before processing the data of Table 3 in the manner illustrated by Figure 7, the simulation data were edited to reduce the number of star sightings to three per vehicle revolution. The relative angular spadngs of the stars transited in the simulation are shown in Figure 8. The three well placed stars (48, 154, 239), all with angular separations of approximately 120 degrees, were selected for this investigation. Seventeen sightings of each star TABLE 4 STAR CATALOG INPUT DATA FOR SIMULATED STAR TRANSITS S ta r Boss Apparent S p ectral Right Declination nunfcer2 catalog Name Cons te lla tio n v isu a l class ascension 6 (degree) number magnitude a (degree) 23 33584 Acamar 0 E ridani 3.06 A2 44.248049 -40.432590 48 6668 B e lla trix y Orionis 1.70 B2 80.828788 6.323596 65 8208 Tejat Posterior y Geminorum 3.19 M3 95.227542 22.534336 102 13157 0 Ursae M ajoris 3.26 F6 142.647602 51.833867 113 15145 Merak B Ursae Majoris 2.44 A0 164.949841 56.566318 154 19607 Haris y Boo 3.00 A7 217.680616 38.484420 172 20947 Alphecca a Corona Borealis 2.31 A0 233.316717 26.829354 219 25180 Kaus B orealis X Sagittaril 2.94 K0 276.478207 -25.443715 225 326161 A scella p Sagittaril 2.71 A2 285.123589 -29.931678 239 28682 a Ind 3.21 G2 308.759045 -47.411104 2Arbritrary Identification number 3Double star 00oo 89 225 •172 239 • 113 11 48 Figure 8. Relative Azimuthal Positions of Stare Transited in Simulation. 90 were used to Insure a data period In excess of a precession period. Precession as used here defines the rotation of the symmetry axis ^ A (kg) about the fixed direction of the angular momentum vector (k^) In apace coordinates with the angular frequency $. The simulated star transit time data used in this study are listed In Table 5. As discussed In Appendix C the transit time errors were assumed to have a uniform probability density function with zero mean, limits of approximately +33 microseconds, and a standard deviation of approximately 19 microseconds. In the simulation studies, errors having a standard deviation of approximately this value were added to the simulated transit time data, through selection, with equal proba bility, error values of +26, 0, or -26 microseconds. These discrete error values were selected to correspond more nearly with the errors expected to be produced by the correlation technique used in the detection of signals generated by a star transit of either coded slit group of the star mapper reticle. A discussion of these discrete time errors is given in Appendix C. Before pro ceeding with the ensenble size study, the estimation algorithm was tested for computational errors. This was done by adding zero errors to all simulated transit time data and examining the estimated state values for this zero "noise" case. The results of this estimatibn are -5 shown below for a value of 3x10 for the convergence factor M: ? - .310758° # - 76.462937° ^ - 51.081052° 6 - 54.126670° * - 316.572700° e*. - .064160° 0 J. 1 - 287.844971 deg/sec e2 - .031017° T - 19.137575 deg/sec TABLE 5 SIMULATED STAR TRANSITS S ta r Vertical slit Slanted slit S tar Vertical slit Slanted slit nuizber transit tine transit time number transit time transit time 48 .0000990002 .0100056644 48 10.5544418112 10.5655394825 154 .3908558561 .3949827371 154 10.9451455045 . 10.9501307238 239 .7749453601 .7864069123 239 11.3292885497 11.3389078107 48 1.1728053894 1.1829952544 48 11.7271469651 11.7378941782 154 1.5635551670 1.5673557742 154 12.1178605504 12.1230839643 239 1.9476484639 1.9590632860 239 12.5019941136 12.5117931882 48 2.3455121239 2.3560535139 48 12.8998527488 12.9102333690 154 2.7362498897 2.7398142828 154 13.2905772118 13.2959134195 239 3.1203519949 3.1315843228 239 13.6747003648 13.6847882628 48 3.5182185844 3.5291265385 48 14.0725586111 14.0826119809 154 3.9089416752 3.9123957581 154 14.4632941037 14.4686006850 239 4.2930564223 4.3039981657 239 14.8474070261 14.8578489943 48 4.6909241183 4.7021581054 48 15.2452641696 15.2550791717 154 5.0816329108 5.0851193709 154 15.6360098860 15.6411489502 239 5.4657617384 5.4763488137 239 16.0201134771 16.0309210901 48 5.8636283988 5.8750984725 48 16.4179694235 16.4276710582 154 6.2543261423 6.2579828010 154 16.8087232066 16.8135822296 ' 239 6.6384675224 6.6486895247 239 17.1928190711 17.2039484731 48 7.0363316256 7.0479120813 48 17.5906747130 17.6004051959 154 7.4270234275 7.4309623065 154 17.9814327500 17.9859415712 ! 239 7.8111732304 7.8210750030 239 18.3655234903 18.3768820557 I 48 8.2090344505 8.2205828990 48 18.7633804555 18.7732776803 ( 154 8.5997258725 8.6040161821 154 19.1541374774 19.1582789107 ! 239 8.9838785336 8.9935534010 239 19.5382269353 19.5496873318 ! 48 9.3817376630 9.3931166137 L 154 9.7724335088 9.7770911582 r 239 10.1565835056 - 10.1661591789 i ; [ r- 92 Comparison of these estimated values with the original state values ♦ of Table 2 shows agreement for all states except for » $Q» and ifi, and these states disagree only in the sixth decimal place or in units of 10”® degree. This disagreement was considered negligible and computational errors were considered to have no effect on the results of the transit time error studies. An ensemble of pointing direction error time histories, Cp(t), was generated for the error sequence study with e£(t) determined from equation 6.17. The superscript on tp denotes the pointing direction error time history resulting from the jth error sequence. Mean values and standard deviations of tp were computed across the ensemble (at the times of the simulated star transits) for ensemble sizes of 10, 15, 30, and 54 e rro r time h is to rie s . The means and standard deviations of cp(tj) were calculated according to l N i Ep(ti) - ^ ^ ep^ti) (6.18) > 1 and N "P2< V - K £ (C p fV “ e p (t±) ) 2 . (6.19) 1-1 * A summary of the standard deviation of the pointing direction error resulting from 10, 15, 30, and 54 error sequences is given in Table 6. As Indicated in this table, worst case standard deviations were selected for stars number 48, 154,'and 239. Bounds on these standard deviations were computed for a 99Z confidence Interval and are shown plotted as a 93 TABLE 6 DEVIATION OF POINTING DIRECTION ERROR OF L AXIS FOR 10, 15, 30 AND 54 ERROR SEQUENCES Standard Deviation of Pointing Direction Error of lg Axla 10 Sequences 15 Sequences 30 Sequences 54 Sequence -4 48 8.358 x 10"4 8.419 x 10~4 8.448 x 10**4 7.751 x 10 154 8.645 8.462 8.691 8.692 239 12.385 12.634 12.157 11.426 48 9.000 8.619 6.552 7.966 154 8.206 9.200 8.838 8.502 239 10.687 11.311 11.528 10.833 48 9.112 9.215 8.596 8.041 154 7.885 7.743 7.718 8.332 239 10.543 11.720 11.661 11.210 48 8.779 8.734 8.385 7.938 154 6.014 6.541 7.240 8.096 239 10.326 10.792 10.928 10.354 48 8.092 8.273 8.067 7.746 154 7.431 7.628 7.948 7.859 239 12.109 11.400 10.450 10.543 48 6.270 8.443 8.617 8.087 154 7.375 8.033 7.919 8.443 239 12.442 11.366 11.747 10.947 48 7.292 8.118 8.286 8.062 154 5.619 7.153 8.053 8.296 239 9.700 9.119 9.352 10.309 48 8.425 8.595 8.407 8.154 154 6.097 5.827 7.192 7.920 239 10.237 10.907 10.595 9.821 48 8.727 8.430 8.396 8.233 154 9.205 8.656 8.492 8.742 239 10.005 9.823 9.600 9.520 48 7.530 7.817 8.236 7.906 154 6.624 7.512 7.482 8.123 239 8.635 10.788 10.555 9.790 48 7.279 7.077 7.759 7.707 154 5.787 6.078 7.132 7.423 239 10.606 11.184 10.254 10.282 48 8.345 8.581 8.446 7.866 154 10.302 9.011 9.101 8.976 239 10.779 9.596 9.950 10.008 48 7.200 7.589 7.872 7.806 154 8.849 8.623 8.528 8.064 239 10.700 10.876 11.219 10.693 SB. 94 TABLE 6.— Continued. A Star Sighting Standard Deviation of Pointing Direction Error of ig Axis Event 10 Sequences 15 Sequences 30 Sequences 54 Sequences 48 7.167 x 10"4 8.171 x 10~4 8.040 x 10-4 7.631 x 10"' 154 4.886 6.783 8.099 8.075 239 9.995 10.581 10.259 10.312 48 7.493 7.924 7.923 7.402 154* 10.444 9.637 9.203 8.387 239 10.937 11.590 10.979 10.551 48 8.264 8.043 7.718 7.461 154 8.347 8.479 9.350 8.892 239 11.140 10.737 11.123 10.846 48 7.460 8.564 8.314 7.887 154 8.487 8.309 8.079 8.458 239 12.124 12.056 10.883 10.951 *Worst cases 95 function of the number of error or noise sequences in Figures 9, 10, and 11. These bounds were determined through application of the Chi Squared test, for establishing confidence limits on standard deviations obtained from limited samples [ref. 31], to the indicated maximum standard devia- tion values of Table 6. A summary of the selected statistics of the errors in the estimates of X resulting from the 10, 15, 30, and 54 error sequences is given in Table 7. It may be noted that statistics for the e rro rs fo r the param eters $Q, dently but were determined for + ipo and $ + co 30. No attem pt was made to relate the errors of Table 7 to the pointing g direction errors of Table 6. However, Appendix G does present a development of the relation ship between pointing direction error and errors lJ The data of Table 7 do illustrate the relative size of the errors in ? for acceptable solutions of pointing direction error as Indicated Ln Table 6. Pointing direction error bounds are given in Figures 9, 10, Lnd 11. As indicated by the plots of these figures, the worst case for tt tie bounds on PpCt^) occurs in the case of ten error sequences. These b: ounds for a l l three stars lie within lim its of 0.003 and 0.0005 degree, Since the goal of the attitude determination procedure was to achieve an accuracy of approximately 0.006 to 0.01 degree, or approximately 22 to 36 arc second*, the case of ten error sequences v s b considered to be adequate fo r the Monte Carlo study. Attitude Determination Procedure Error An aly sls This section summarizes the results of an erro c analysis conducted to determine the accuracy of the attitude determination procedure in terms of statistics of the pointing direction error (see equation 6.17) Standard Deviation of 1B Axis Pointing Direction Error (Deg) 10 20 30 x 10 x 30 0 0 iue . 9 Cniec Bud o Sadr Dvain f 1- of Deviation Standard on Bounds Confidence 992 9. Figure -4 10 xs onig rcin rr Str 8vrt case). 48-vorst tar (S rror E irection D Pointing Axis ubr f rr Sequences rror E of Number 20 oe Bound Lover 30 i lto Results R ulation Sim pe Bound Upper 40 * A 96 6050 Standard Deviation of Axis Pointing Direction Error (Deg) 0 10 x iue 0 99 10. Figure 10 xe onig rcin ro ( a 14wrt e). se a c 154-worst tar (S Error irection D Pointing Axle X ofdne ons n tnad eito o ig of Deviation Standard on Bounds Confidence ubr f ro Sequences Error of Number 20 050 30 esults R ulation Sim oe Bound Lower pe Bound Upper 60 97 ! I i Standard Deviation of 1. Axis Pointing Direction Error (Deg) 30 x 10“4 x 30 10 20 01 “ - 0 iue 11 Figure 10 xs onig rcin rr Str 3*os case). 239*«orst tar (S rror E irection D Pointing Axis 99 X Confidence Bounds on Standard Deviation of ig ig of Deviation Standard on Bounds 99 X Confidence ubr f ro Sequences Error of Nuober 20 30 i lto Results R ulation Sim 05 60 50 A0 oe Bound Lower pe Bound Upper , * 98 99 TABLE 7 STATISTICS OF THE ERRORS IN X (DEGREES) Parameter 10 Sequences 15 Sequenc as Mean Standard Mean Stan lard Deviation Devi atlon .00014 .00119 .00005 .00120 .00063 .00257 .00012 .00244 .00012 .00340 .00015 .00327 -.00019 .00165 .00008 .00163 '< * 0 + *„> -.00001 .00009 .00000 .00 Oil w. -.00841 .05761 -.00357 .06206 .00046 .00303 .00020 .00 337 Parameter 30 Sequences 54 Sequences Mean Standard Mean Stan lard Deviation Devi itio n .00009 .00126 .00009 .00131 e0 * -.00004 .00259 .00023 .00282 e0 .00036 .00290 .00017 .00!258 X + *o> .00020 .00194 .00003 .00221 .00001 .00010 .00001 .00010 s -.00833 .06230 .00034 .05 >98 s .00055 .00351 .00020 .00300 % A 100 of the opltcal axis (lg)< The variation of this error was Investigated as a function of the data fitting period (In terms of fractional parts of a precession cycle) and as a function of the number of transited stars per spin period. The effects of star placement on the pointing direction error was also examined. It should be mentioned here that the block diagram of Figure 7 also applies to the study of this section, except that the number of sequences of noise or measurement errors added to the simulated transit times Is set at ten for all variations of this study. The values of )£, e, and I. listed In Table 2 1 V V 8 8 J were also used In the simulation of the star transit time data for this section. In the simulation of this section, transit times were also compiled in a listing like that of Table 5. However, for this error analysis, simulated transit times were computed for all the stars of Table 4 and Figure 8 instead of only for the star set (48, 154, 239) as done for the earlier ensemble size study. In addition, transit times were compiled for thirty seconds so that pointing direction errors could be determined for data fitting periods of one-half, one, and one and one-half precession cycles. For each data fitting period, right ascension, declination, and pointing direction errors were determined for two, three, four and six star transits per vehicle spin period. It Is also noted from Figure 7 that ten noise or transit time error sequences were added to each sim ulated Bample of s ta r d ata. Thus ten runs through the least squares program were necessary to determine the resulting ten estimates of the original nine parameters. Each of the ten sets of nine parameters was then fed into a time history generator, i.e., the assumed vehicle attitude equations were written as a function 101 of time. Beginning at time zero, and for equal Intervals of time, time histories of attitude data for the star mapper optical axis were calcu lated fo r each s e t of nine unknowns. Each of these time h is to rie s was computed for the length of time covered by the selected sample size of simulated data used in the least squares program. This provided time history records of attitude or pointing direction both at the times of star sightings and during the periods of time between star sightings. The attitude data representing times between star sightings provide a measure of the interpolation capability of the attitude determination procedure. Each of these ten time histories was compared to a reference time history generated by the original nine parameters used in the simulation program. This comparison resulted in an ensemble consisting of ten error time histories in right ascension, declination and in A pointing direction for the ig axes for each combination of two, three, four or six star sightings per revolution of the principal spin axis and data fitting periods of one-half, one, and one and one-half preces sion periods. Thus, for each of the three errors examined (ea, and £p), there were twelve ensembles, each consisting of ten error time histories representing the particular error parameter of interest. In addition, two special cases of poorly located stars were examined. These special cases were for two star transits per vehicle spin period over a data fitting period of one-half precession period, and three star transits per vehicle spin period over a data fitting period of one precession period. Each time history of the error in right ascension (e^) and the error in declination (c^) was determined through use of equations 6.3, 6.4, 6.8 and 6.9 for the ig axis. A time history of the pointing direction error (ep) for each member of an ensemble was calculated using equation 6.17. Means and standard deviations of these errors were calculated using equations 6.18 and 6.19. Statistics of the errors in right ascension, declination and pointing direction for ig were calculated across their ensembles at specific instants of time t^. The designations for the standard dev iation of these errors are atM* ^^1^18* ^ the earlier ensemble size study, the maximum value of standard deviation computed acrosB the ensemble was selected as the value of interest. Spacing of the time points was selected to be 0.01 second. Thus each ensemble of ten error time histories consisted of approximately 1000 to 3000 computed data points for the one-half and one and one-half pre cession period cases respectively. Several members of each ensemble for eQ, Cg, and ep were selected and root-mean-square values were com puted for each of these error time histories. No attempt was made to correlate root-mean-square values of specific errors to the standard deviatiops of the same error. However, it may be noted that the maxi mum value of standard deviations computed across an ensemble of errors agreed with the root-mean-square value of this error to within approxi mately 15%. The stars of Table 4 and Figure 8 selected for this error study are listed in Table 8. Stars 23 and 225, selected for the two stars per spin period, are separated by 90% and stars 48, 172 and 239, used for the three stars per spin period case, are separated by angles of 100° to 137°. The stars selected for the cases of four and six stars per spin period are distributed throughout a complete spin cycle. 103 TABLE 8 SELECTED STAR GROUPINGS FOR THE ERROR STUDY Star/Spin Period S tar Identification 2 23, 225 00 3 .c* 172, 239 4 65, 102, 113, 219 6 65, 102, 113, 219, 225, 23 A summary of the results of the error study using the stars of Table 8 Is Illustrated in Figures 12, 13 and 14. In these figures, the maximum values of 0o(t 1)£g» °6^ti^i8 and ap^i^£8 are Plotted aa a function of the number of transited stars per spin period and time duration of the data fitting period In terms of multiples of a preces sion period. As indicated in Figures 12, 13, and 14, the accuracy of attitude determination increased as the data fitting period used in the least-squares program increased and as the number of star transits per spin period increased for the one precession cycle case. The one- half precession cycle case also indicates evidence of this trend. The one and one-half precession cycle case shows no p a rtic u la r trend and the results for this case were interpreted to mean that the errors in the estimates of X were in the noise level for any considered number of stars per spin period. For all three cases of data fitting periods, the observed maximum value of apCfc^)£g was ^eaa than the maximum value of 0.01° as specified for the accuracy goal for the attitude determination procedure. The values of Op(t^)*g of Figure 14 were subjected to the Chi Squared test in order to establish the bounds or lim its of the true Maximum Standard Deviation, o Degree Figure 12 Figure (X 1 I B 10 7 - 4 oL - 5 * 3 1 “ 1 2 - l . Maximum Standard Deviation in Right Ascension of ig Axis vs. vs. Axis ig of Ascension Right in Deviation Maximum Standard . ubr f astd as e Si Period. Spin per tars S ransited T of Number 4 3 5 6 5 4 3 2 ______astd tr Pr pn Period Spin Per Stars ransited T I ______I ______■ 0 1/2 Precession Precession 1/2 ■ 0 4 1 Precession Precession 1 4 ) 12 Precession 1/2 1 6) Period Period Period I ______n I Figure 13. Maximus Standard D eviation in D eclination of ig Axis vs. vs. Axis ig of eclination D in eviation D Standard Maximus 13. Figure O Maximum Standard D eviation •o oo 7 x 10 x 7 2 - -3 rfe o Trnie Str pr pn Period. Spin per tars S ransited T of Nrafcer astd tr Pr pn Period Spin Per Stars ransited T s Precession 1 $ Q 1/2 Precaaalon Precaaalon 1/2 Q 1-1/2 Precession Precession 1-1/2 Period Period Period 105 Maximum Standard Deviation, Degree 0 3 10“ 7x 0 1 2 4 - 3 - 5 6 Figure 14 Figure - 1 . Maximum Standard Deviation of Pointing D irection irection D Pointing of Deviation Maximum Standard . e Si Period. Spin per E rror fo r lg Axis vs. Number of T ransited S tars tars S ransited T of Number vs. Axis lg r fo rror E J 3 5 6 5 4 3 2 ______astd as e Si Period Spin Per tars S ransited T I ______I ______(5 1-1/2 Precession Precession 1-1/2 (5 © 1/2 Precession Precession 1/2 © £ 1 Precession Precession 1 £ Period Period Period ■ i 106 107 value of maximum ap(fcj)£g with a confidence factor of 99%. Since ten time histories or records constituted the size of the ensembles used In this error analysis, the multiplying factors for the observed maximum values of 9p(t£)£g are 0.616 and 2.278. These multiplying factors were applied to the data of Figure 14 and the re s u lts a re shown p lo tte d in Figures 15, 16, and 17 for the one-half, one, and one and one-half pre cession period cases respectively. These figures Indicate that there are sets of conditions for which the upper bound of the 99% confidence Interval will exceed the design goal limit of 0.01°. The conditions noted from these figures are: a data fitting period of one-half pre cession period and three stars per spin period, and a data fitting period of one precession period and two stars per spin period. Again the case of a data fitting period of one and one-half precession periods appears to give results which are within the noise level for any number of stars per spin period. The results plotted in Figures 15, 16 and 17 were factors in the selection of star transit frequency and data fitting periods used in flight data reduction and analysis. Other factors used in these selections are star placement within a spin period and inter pretation of the time residuals of the least squares estimation proce dure. The latter factor will be discussed in the following section and in the chapter dealing with flight results. The e ffe c t of s ta r placement on p o inting d ire c tio n e rro r was examined and the results are presented in this section. The conditions considered for this examination are given in Table 9. For each of the two cases of this table, ten transit time error sequences were used to i produce ten estimates of X and therefore ten error time histories or Figure 15. Figure Maximum Standard Deviation, o (t.)" , Degree p p 1 10 14 16. Maximum Standard D eviation of Pointing D irection E rro r, r, rro E irection D Pointing of eviation D Maximum Standard 16. 10 x 2 4 6 5 4 3 2 1 Confidence Bounds v s. Nuafcer of S ta r T ran sits per Spin Period (1 Precession Precession (1 Period Spin per sits ran T r ta S of Nuafcer eriod). P s. v Bounds Confidence .-3 1 ubr f as e Si Period Spin Per tars S of Number 1 ______oe Bound Lower i i lto Results R ulation Sim pe Bound Upper a xO 1 P wih 99X ith w , a c ) . t ( I w o iue 7 Mxmm tnad eito o Pitn Dieto Eror a(f£* t 99Z ith w ap(tf)£g* r, rro E irection D Pointing of Deviation Maximum Standard 17. Figure Maximum Standard Deviation, Degree 6 10“3 x16 1 ofdne ons s Nscr f ar a is e Si Pro (-/ Pre (1-1/2 Period Spin per sits ran s). T d r ta erio S P of Nusfcer cession vs. Bounds Confidence 2 astd as e Si Period Spin Per tars S ransited T 3 4 5 i lto Results R ulation Sim pe Bound Upper 6 oe Bound Lower 110 TABLE 9 ATTITUDE DETERMINATION ACCURACY FOR THE CASE OF POORLY PLACED STARS Selected Angular Data Fitting “ax W i e 99Z Bounds S tars Separation Period 65 i(65, 219)-177° 1/2 Precession 0.0767° (.0472°,.1747°) 219 Period 65 £(65, 102)«46° 1 Precession 0.0420° (.0259°,.0957°) 102 £(65, 113)«60° Period 113 ^(102, 113)-40° 112 records of cp. Standard deviations of ep were computed as described earlier with the maximum value of crp listed In Table 9. The 99Z confidence bounds were computed using the multiplying factors of the Chi Squared test for ten samples* The data of Table 9, especially when compared to the data of Figures 15, 16 and 17, show that star placement within a spin period Is a major factor in the accuracy of pointing direction determinations. It should be noted that the two stars per spin period are separated by an ambiguous angle of nearly 180° and the three stars per spin period case are all located within a 90° sector of a spin cycle. With the poor placements of the stars of Table 9, precession information Is not well defined. This examination of star placement within a spin period resulted In a requirement that for the case of flight data the stars selected for least squares processing would be distributed throughout a spin cycle. Based on the results of Figures 15, 16 and 17 and the exclusion of poorly placed stars, the data fitting period for flight data was selected to have a lower limit of 3/4 of a precession cycle and the data of this period were required to contain at least 4 to 6 star transits per spin period. These requirements, along with the results of the error studies, were considered to give assurance that the 'accuracy of the pointing direction computations arising from flight data would meet the design goal of 0.01° with a confidence level of 99%. This assumption is based on the hypothesis that the errors In time measure ments experienced In flight and the mismatch of the dynamical models of the flight vehicle and estimator will be approximately the same as 113 those assumed during the simulation studies. Differences In these two factors experienced during flight data evaluation, w ill be discussed In the chapter dealing with flight results. Relationship of Transit Tima Errors to Least Squares Functional Residuals and a Method for Assessing the Adequacy of an Estimate The previous sections of this chapter presented the results of studies of the accuracy of attitude determination resulting from the use of estimates of X as computed through use of the least squares estimation procedure. In these studies, transit time errors were added to the simulated star transit times t^. The values of Eti had an equal probability of having values of +26, 0, or -26 microseconds and a standard deviation ofc of approximately 21 microseconds. Since It Is not possible to measure a for flight data, it was necessary that some other parameter be used to estimate the magnitude of transit time errors. As there Is no attitude reference by which the accuracy of the attitude determination procedure, when applied to flight data, may be Independently checked, the only accuracy numbers which are available are those generated by the estimation procedure Itself—that Is, the computed mean and standard deviation of the solution residuals, which In this case are estimates of the error In the observation times as described In Chapter 5. However, it must be noted that for the case of flight data, these errors reflect not only the input stellar noise and system noises, but uncertainties due to the finite resolution of the optics, and the fit of the observed data to the assumed theoretical dynamical model of the estimator which w ill probably not be In perfect agreement with the real dynamical model. 114 The adequacy of any estimate of X may be determined by the use of certain assumptions. If Is assumed to be a good enough estimate '*“(1+1) t to allow the use of X and the true ith transit time t ^ to satisfy approximately the constraint of equation 3.18 then JjDEHR tt 0 (6 . 20) For convenience, let the functional dependence of equation 6.20 on X and t for any star and the (j+l)st estimate be denoted by J2DEHR - f ^ (j+1), tt1 ) (6 . 21) Equation 6.21 may be approximated by a Taylor series expansion djout the tlme^t^ - A t^ as (6. 22) At, By defining the quantity ^t*^ - At^Jto be t . - t - At. (6.23) ml 1 1 where t ^ 1 b any measured transit time, and by setting fO T,^) equal to zero by use of equation 6.20, the time error In the observed ith transit time may be approximated by 115 At^ « - (6.24) 3£^ |(3 + 1>> ^ t - t'ml since this quantity At^ has been shown, In Chapter 3, to be related to by the expression (3.21) In previous error studies of this chapter the convergence factor M was adjusted until a least-squares Iteration resulted In time residual values At^ which agreed with the magnitude of to within an accuracy of 5 microseconds. In these studies the standard deviation of At^ was calculated for each set of k stars used in a least-squared solution. This standard deviation of the time residuals microseconds, the accuracy of the computed vehicle-attitude orientation was well within the specified requirements. This parameter, c , was the factor which was used to assess the adequacy of a least-squares solution for flight data. In the case of flight data, the convergence factor M was adjusted until a. was near i t s minimum value. When as an adequate one. Additional uses of these time residuals and their statistics in the interpretation of flight data are discussed In the next chapter, which deals,with flight results. CHAPTER 7 FLIGHT RESULTS Scar mappers of the type described In this dissertation were flown on two NASA sponsored sub-orbital flights of vehicles designated as Project Scanner Spacecraft. The spacecraft, attitude control system and trajectory are briefly described In this chapter. A brief discus sion of the star mapper and data processing procedures Is also presented. Additional description of the transit time detection technique and the star mapper design may be found in Appendixes 8 and C. Flight results are summarized in the form of listings of stars that were detected and identified. Estimates of X developed through processing of these identified stars are presented and pointing direction computations re sulting from these estimates are discussed. Signal and noise character istics observed in flight are compared to predicted characteristics and anomalies noted in the flig h t d ata and computed a ttitu d e data are d is cussed and related to errors in the assumed dynamical model. Spacecraft and Launch Data A brief description of the spacecraft, which was built by Honeywell Corporation, is given in Table 10. Both flights were launched at Wallops * Island, Virginia, and the nominal trajectory for both flights was de signed to be along an azimuth of approximately 67° southeast of the launch site. The nominal Impact point was selected to be 700 nautical miles down range and approximately 25° south of Bermuda. The planned trajectory dispersions were approximately +165 and -220 nautical miles 116 117 in range and +16° in azimuth. An elevation profile of the nominal tra jectory and allowable range and altitude dispersions are illustrated in Figure 18. This figure also indicates points of significant time events. An attitude control subsystem alined the longitudinal axis of the space craft to a local vertical four times during the flights with the aline- ment periods pre-programmed prior to launch. After burnout of the X248 rocket motor, two cables, each with a weight at one end, were allowed to unwind from the X248. These weighted cables reduced the spin of the spacecraft to a value within a range of 0.65 to 0.85 revolutions per second. The research package consisted of a star mapper built by Baird Atomic Corporation and two earth scanning radiometers, built by Santa Barbara Research Corporation, which were used to obtain measurements of the Infrared characteristics of the earth's atmosphere near the horizon. Values of the spacecraft moments of inertia for the first flight vehicle are listed in Table 11. TABLE 10 SCANNER SPACECRAFT Vehicle: Modified Trailblazer II booster configuration - 3 stages: 1 st A erojet - General 28KS-57000 (Junior) with two Thlokol XM 19El (Recruit) motors attached for thrust augmentation. 2nd Thiokol TX77-7 (Skat), and 3rd - X248 A-6 (Altalr). Payload weight: 560 pounds (Includes burned-out X248 motor) Payload size: Diameter - 2 feet Length - 12 fe e t Altitude, feet rX-7 gnii . n itio n Ig -77 |rTX n 2 10 x 3 1 Mf’SX-ll p U Erec tio n 1 n tio 1 st st 1 28 gnii n itio n Ig X248 rection E 28 Burnout X248 Burnout Burnout 2nd Despin 1 iue 8 Nmnl n + o aetr Pr ies. s file ro P rajectory T 3o + and Nominal 18. Figure r Erection E 3rd 4th 2 Down Range D istance, fe e t t e fe istance, D Down Range 4th 3 t Erection E 4th 4 ATT 119 TABLE 11 SPACECRAFT MOMENTS OF INERTIA Configuration *3 2 *1 2 (S lu g -ft ) (S lu g -ft ) Launch 328 64100 Recruit Burnout 455 60300 1st Stage Separation 263 40900 2nd Stage Ig n itio n 40.2 5396 2nd Stage Burnout 31.95 3592 Spacecraft Separation and X248 Ig n itio n 11.04 221.8 X248 Burnout 8.27* 130.34* *Estimates It should be noted that the Inertia values at X248 burnout are estimates only due to the uncertainty of total burnout of fuel. It should also be noted that uneven burnout of X248 fuel Is the major contributor to an Imbalance of the inertias 1^ and I^. As indicated In Chapter 2, an allowance of 10Z haB been specified for a mismatch of these two inertias. This mismatch of 10Z In 1^ and Ig was also used In the generation of transit time data In the simulation studies. It should be noted that this mismatch percentage could be a much smaller value for an orbital vehicle since all rocket stages would be disengaged from the final pay load and the moments of inertia of the payload could be measured to within 0.1Z prior to launch. The dynamical model of the estimator, used in the processing of flight data, Is assumed to be that of a symmetrical 120 vehicle as was also assumed in Che simulation studies. During both flights, a cold gas reaction jet control system, of the type discussed in Chapter 2, was used to erect the spacecraft to within +2° of a local vertical at four specific times or locations along the trajectory. During the time intervals between these erection or control times, the control system was turned off. These time intervals are referred toas star mapper data periods during which the spacecraft was assumed to be free of external and control torques. These attitude control system (ACS) and data time intervals were preset before launch as listed in Table 12. From this table, it is seen that the time dura tion of the 1st, 2nd, 3rd and 4th control periods are 45, 20, 20 and 20 seconds, respectively. Also, the time duration of the 1st, 2nd, 3rd and 4th data periods In seconds are 141, 130, 150 and unspecified, res pectively. Flight data, consisting of star mapper information used for vehicle attitude determination, were confined to these four data periods. TABLE 12 CONTROL SYSTEM AND DATA INTERVAL SCHEDULE Time from lift off Events (seconds) 154 1 st ACS A ctivation 199 1 st ACS Shutdown 340 2nd ACS A ctivation 360 2nd ACS Shutdown 490 3rd ACS A ctivation 510 3rd ACS Shutdown 660 4th ACS Activation 680 4th ACS Shutdown 121 A comprehensive description of the operation of the class of attitude control system used on these flights may be found in Reference 32. However, for completeness, a brief description w ill be presented here. Basically, the control system is required to control the princi pal spin axis of the spacecraft to the local vertical and to damp out A A A angular rates about ig and jg. Deviations of the kg axis from the local vertical are sensed through use of a horizon telescope. This telescope, which is body mounted, utilizes the spacecraft spin for its scan of the earth's horizon, which in turn provides a measure of the attitude error. An illustration of this concept is shown in Figure 19. As seen in this figure, the telescope is set in the spacecraft at such an angle that when the spacecraft spin axis is alined with a local vertical, the earth's horizon falls within the field-of-view of the telescope. As the spacecraft spins in this attitude, the same percent age of the telescope's field-of-vlew is filled by the earth's horizon A at all times, and hence no attitude error is sensed. If k, is not 0 alined to the local vertical, the percentage of the telescope field-of- view occupied by the earth's horizon will vary as the spacecraft spins. This variation sensed as an attitude error will cause the attitude control system to be activated. This control system, which is a cold A gas reaction jet type, will cause the kg axis to preceBS in such' a direction as to cause the attitude error to be reduced. Operation of this control system during the specified control periods is designed to maintain alinement of the kg axis to within +2° of the local vertical. The effects of improper operation of this control system on the star mapper data and estimation results w ill be discussed later in this chapter. 122 Center of mess Looj^ angle of telescope Thrust1 vector Vehicle ground path Figure 19. Illustration of Control Action of the Attitude Control System. The measurements used to obtain attitude Information were made with a passively scanning star telescope. The telescope is shown mounted in the spacecraft In Figure 20. The telescope with a field-of-view of 6° by 6*, is mounted with its optical axis normal to the expected principal spin axis of the spacecraft. The star telescope is briefly described here and a more extensive discussion of the design considerations of such a telescope may be found in Appendix C. The coded r e tic le , shown in Figure 21, was centered in the focal plane of the telescope, and a photo detector was placed behind the reticle. The reticle was princi pally opaque and contained a vertical and slanted set of coded trans parent slits with the narrowest slit of each set being 0.015°. The coding feature of the reticle was used to reduce system response to noise and to the multitude of stars dimmer than a selected threshold visual magnitude. This coding feature was selected to insure a minimum of three detectable and identifiable stars per spin period (see Appendix B). As illustrated in Figure 21, spin motion of the spacecraft causes star images to travel across the reticle producing two coded sequences of radiant energy which are sensed by the photo detector and then transm itted to a ground receiving s ta tio n . The ground received coded star signals are tane recorded for processing at a later time along with range time signals having a resolution of ten microseconds. ' Pro cessing of these recorded signals consists of four significant opera tions: (1) time detection of apparent star transit signals, (2) sorting of the apparent transit times to reduce false alarms, (3) identification of the detected stars, and (4) determination of the spacecraft attitude time history. The procedures used for signal and data processing will be briefly described in the next section. Principal spin axis Local v e rtic a l Star mapper Field of view of horizon telescope Earth Radar tracking Real-tim e data telemetry 124 Figure 20. Spacecraft Operational Sketch. Star action « ★ Star 2 _nojuLStar 1 ii ^ + 1 * **-1 Figure 21. Illustration of Reticle and Pulse Groups Generated by Stars 1 and 2. (Note: Lines In Reticle Sketch Represent Transparent Slits.) Ulro 126 Star Mapper Signal and Data Processing Procedures As Illustrated In Figure 22, the recorded star mapper signals along with range time were processed by a pulse-code detector that Identified the transit times for each detected star signal. A variable threshold control was adjusted to lim it the number of detected s ta r sig n a ls, and the peak amplitude of the first pulse In the sequence of pulses gener ated by a star was recorded for purposes of classifying the stars according to their signal levels. The processing required to detect star signals is briefly described here. Further discussion may be found in Appendixes B and C. As shown In Figure 22, the recorded coded star mapper signals were compared with a threshold level and all signals exceeding this threshold were con verted to a digital signal with a star presence in a slit designated by a one and a star absence In a slit designated by a zero. A single pulse sequence for a particular star transit consisted of a 9-bit code pattern. The code pattern produced by the reticle slit design was 1—1—0—1—0-0-0—1-0. The time duration T of a star signal generated by a star transit across the narrowest transparent slit was set by T » U) — e a t * * 9 where p is 0.015° for each element of the nine-element code and the estimated spin rate m was determined from examination of an oacillo- es c graph record of the raw star mapper data in which time periods between repeated sightings of the more prominent star signals were assumed to be the time period for a complete vehicle spin cycle. Estlmted Teleaetery Analog to receiver and Star algnal level, tape recorder digital converter transit tines Variable Decoding logic Listing of atara: Star Manual aortlng of I Star ww*i«r Identification atar tranalt tine > Vise of algbtlnga Reduced atar A p rio ri Catalog of updated catalog data atar epbenerla A p rio ri data Identified parmaeter Convergence Convergence yea te a t of dynanlcal te a t > equations of notion no no Figure 22. S tar Mapper Data Processing Procedure. 128 The sample time used In the analog-to-digital converter was select ed to be T/4 in microseconds. The presence of a star in a single element was Indicated when two or more of four consecutive sample time in te rv a ls had signal levels which exceeded the variable threshold level. The un certainty in the time of occurrence of a star presence or a star absence signal was +T/2 microseconds. The output of the analog-to-digital con verter was cross-correlated in the decoder with a prestored replica of a code group as illustrated in Figure 22. A star transit time t^ was acknowledged when correlation of the reference code sequence with any nine-element coded sequence produced agreement of at least two of four ones and all five zeros or at least three of four ones and at least four of five zeros. The transit times of apparent stars were recorded T/4 microseconds after the time of occurrence of the trailing edge of the ninth code element. The delay time of T/4 microseconds was necessary for readout of range time. The total uncertainty in star transit time included the +5 microsecond uncertainty in range time and +T/2 micro second uncertainty in the time of occurrence of a star presence or star absence signal. The total time error was conservatively assumed to be the sum of the two errors of + microseconds. This total time error was assumed to have a uniform probability density function and therefore a standard deviation of approximately 19 microseconds for the nominal value of T being approximately 56 microseconds. At this point, a distinction must be made between the physical groups of slits on the actual reticle and the analytical slit edges for which star transit times are defined. A transit time,, recorded by the correlator as the time at which a atar transits some point fixed with 129 respect to the reticle, corresponds to a line on the reticle located 9.25p degrees away from the leading edge of the vertical or slanted groups of slits. These analytical time lines on the reticle are defined to lie in planes designated as the vertical and slanted slit planes, respectively. A geometrical description of these slit planes is dis cussed in Appendix F. The output of the decoder was a listing of apparent star transit times and their signal amplitudes. This listing was hand sorted in an attempt to relate transit times to discrete star sightings. The listing was examined for pairs of coded sequences of approximately equal ampli tude such as a single star's transit of the fleld-of-vlew would be expected to produce. This amplitude sorting was repeated for all vehicle spin cycles as determined from the estimated vehicle spin rate. Additional sorting was done by searching for a recurrence of approxi mately equal time intervals between a pair of pulse sequences produced by the same star and the recurrence of approximately equal time Intervals between pairs of sequences produced by different stars. All transit-time differences that exceeded the constraints established by the geometry of the reticle slit configuration and estimated vehicle spin rate were rejected aB false star indications. ThiB modified listing of star transit times was applied to the star identification procedure described in Appendix D. The final step in the processing of star mapper data is the application of Identified stars and their transit times to the parameter estimation procedure discussed in Chapters 3 and 4, A summary of the received star mapper signals and identified stars of both flights are presented in the following section. 130 Summary of Signal Characteristics and Identified Stars The two flights of the star mappers described In this dissertation were on August 16, 1966, and December 10, 1966. Figure 23 shows a typical scan of the star mapper field-of-view across the celestial sphere during the first flig h t.. This scan was reconstructed by use of two of the Identified stars. Photographs of the coded star signals for the selected stars Alkaid and Alphecca are shown in Plate I. Each photo graph shows the star signals generated by the vertical group of slits for three different sighting times to demonstrate the severe noise problem Introduced by background and signal-generated noise. In all photographs, the coded pattern is clearly evident and indicates the advantages of coded reticles for star-signal detection purposes. Alkaid and Alphecca are +1.91 visual magnitude B3 and +2.31 visual magnitude AO stars with predicted star mapper signal levels of 0.90 and 0.60 volt (see Appendix C) respectively. Several sightings of each star resulted in an average output signal level of 1.65 volts for Alkaid and 0.85 volt for Alphecca. Several other stars were examined in a similar manner and results indicated that the star mapper had a sensitivity which was 1.5 to 1.75 times greater than that expected from the pre- flight calibration results. A check of the star mapper calibration » over its dynamic range was not possible since none of the stars identi fied during the first flight were predicted to produce outputs greater than 1 volt. The average background noise was expected to be approximately equivalent to the signal produced by a +2.8 to +5.0 visual magnitude A0 star. No attempt was made to verify this expected background noise OKllM tton, M g -80 30 •3 90 -9 .40 "SO- 20 GO 0 2 90 - 0 3 TO 10 - - - Figure 23*. Typical Scan of S ta r Mapper F ield of View. Scanner F lig h t 1. 1. t h lig F Scanner View. of ield F Mapper r ta S of Scan Typical 23*. Figure Turns* O O O O 2 10 O O 6 IQ 8 10 0 0 2 190 180 ITQ 160 QO MO 130 120 TO CO 0 9 0 8 TO GO Fas ndi e Str Utlze n Dt Reduction.) Data in ed tiliz U tars S te a ic d In (Flags ITB Right cncantion . d«g . cncantion Right O Z 3 240 20 7 290 290 300 30 3 340 330 360 6 3 0 3 3 0 4 3 330 0 2 3 340 0 0 3 0 9 2 0 9 2 270 0 6 2 250 0 4 2 230 0 2 2 ■n:. ■n:. i u i c k umusis u x n ALKAID ALPHECCA Plate I. Typical Coded Star Signals Received from Alkaid and Alphecca During the First Flight. 133 level throughout the flight. However, a rough approximation of the peak-to-peak noise level due to background noise may be determined through an examination of the photographs of Plate I. In each photo graph, the average peak-to-peak background noise level is seen to be approximately 0.3 volt and the root-mean-square (rms) noise has an approximate value of 0.11 volt. The average in-flight signal of 0.85 volt obtained for Alphecca and the average rms background level of 0.11 volt indicate a slgnal-to-nolse ratio of 8.1. This average signal-to-rms background ratio of 8.1 may be compared to that ratio predicted through use of the signal and noise definitions given in Appendix B and the star mapper design characteristics given in Appendix C. Using equations B.l and B.2 of Appendix B, an assumed +2.3 m^ signal star, an expected of 270° per second, and an assumed worst noise case of one thousand +10 m^ stars per square degree, the slgnal- to-noise ratio is found to be S (+2.3) - 11 (7.1) The quantities +2.3 and +2.82 refer to visual magnitudes of the signal star and equivalent visual magnitude of the background noise. The thermionic noise term defined by equation B.3 of Appendix B is*not included in the expression of equation 7.1 since N_ « 100 N where N_ o X a is the average background noise. It should be noted that the slgnal-to- nolse ratio of equation 7.1 is expressed for the photomultiplier cathode. Thus this ratio will be reduced when expressed for the output or photo multiplier anode. This reduction is due to the multiplicative factor 134 of applied to the rms noise, where k Is the gain per dynode stage. ^ ' J Since the photomultipliers used In the subject flights had 14 dynodes, , 6 - and a selected tuhe gain of 10 , k had a value of approximately 2.7. Application of this spreading factor to equation 7.1 results in a slgnal- A I',(.it ' ■ ’.V i i * ■ n ‘ ' ■!.' I " '' ' ■;•••: ' 1; i..:v \ \ >■< ' v \ r to-nolse ratio of 8.7. This predicted slgnal-to-nolse ratio Is further !r>'! '« \ • ■ 1 ■; ■ . . ■: * ;■ r '••• ! ..«• :h. 1 ' ! n.. reduced by the increased value of experienced In flight. The flight i ■ f ' I . . v 11 - ■■■ : value of 300 degrees per second rather than the predicted 270 degrees T , * per second results In a slgnal-to-nolse ratio of 8.26, which agrees favorably with the value of\8,1 obtained,from flight data. These two ratios and the Increased value of sensitivity may‘be used to determine the relationship of predicted background noise to that experienced In flight. If the Increase in sensitivity ,1s assumed.tobe 1.5 then •, j r * .‘if ; «■ ;; v.-• • i • • . ; i: : \ . S (+2.3) ^ H b (+2.82) 1.5 S (+2.3) I t f • •>f •• .. i- t r-’ •. ■ r ■1 . I i ;' • V 1<5 NB (X) • i ....1 i i! ;■ v •. i : <"i; w \ \ :» i ■; • • • where x denotes the equivalent signal level for the flight background ’ tv • •( : r. ! i f. . ■ I ‘ . Vvii. I •’ ■: 1 i. ■ , * . noise. Then .... , i ■ • ••• , i ’ ■ • ; 1 1 i • •• t -1.56 NB i(f2.82) .. . „ ,„;n, . ,Uv. i , », (7.3) From equation 7J3 the background noise' may be expressed‘as « * an equivalentf * ‘ signal of +2,33 rn^Y This Increased level of background noise"results1 in an Increase'in the expected rms value of background noise.1 ’The amplification factor1 is seen to be ! 1 - 135 ^ 1.5 Nb (x ) ------1*52 . (7.4) ^ Nb ( 2 .8 2 ) A more significant factor is the increase in the predicted rms noise level by\S + Ng since this Is the noise present during the star signal detection process. The factor of Increase for this rms noise is found from ^ 1.5 (S (+2.3) +NB (+2.33) ) ------1.3 . (7.5) y ? (+2.3) + Ng (+2.82) It should be noted that this factor of 1.3 is due to the increase in sensitivity, increase in w^, and increased background noise. This factor of increase in yj S + will be related to estimates of the standard deviation of time measurement errors obtained from flight data in a following section. All values of signal-to-noiBe ratios resulting from flight data are in close agreement with predicted values and lend valid ity to the signal and noise models used in the design of the star mapper. Figure 24 shows a sample oscillographic record of the output of the s ta r mapper. The record shows the sig n a ls generated by the tr a n s its » of two different stars across the vertical and slanted groups of slits. The code groups are degraded somewhat due to the frequency response of the oscillograph. It is noted that one of the stars identified on this trace is a +2.92 visual magnitude A3 star. A listing of all stars identified during the first flight is given in Table 13. These stars are listed according to their Boss catalog number, star name, Bayer Amplitude, volte O 2 1 3 ■I • *-«— iue 4 Osilgah eod f upt Str Mapper. tar S f o Output of Record scillograph O 24. Figure 10 0 E rldani (Acasar) (Acasar) rldani E 0 2.92 - ie msec Time, 20 .,— « ■ « ■ 1 / ^ i «t. ,i—i 50 *1 Y O riools (B e lla trlx ) ) trlx lla e (B riools O Y * 1 * *« y 1.70 - ■y . ■ ■■ ■■ 60 I 136 137 TABLE 13 IDENTIFIED STARS Boss catalog V isual S pectral S tar name Bayer name number magnitude, class % First Flight *3584 Acaroar 6 E ridani 3.06 A2 6274 Curs a 8 E rldanl 2.92 A3 6668 B e lla trlx Y O rlonis 1.70 B2 6847 Mintaka S O rionis 2.48 09 8208 Tejat Posterior V Gemlnorum 3.19 M3 12407 T alith a t Ursae Majorls 3.12 A4 15145 Merak 8 Ursae Majorls 2.44 A1 16268 Phekda Y Ursae Majorls 2.54 A0 17518 Alioth e Ursae Majorls 1.68 A0 *18133 Mlzar C Ursae Majorls 2.17 A2 18643 Alkaid n Ursae Majorls 1.91 B3 20947 Alphecca a Corona Borealis 2.31 A0 22193 Komephoros 8 H erculls 2.81 G8 25180 Kaus Borealis XSaglttarll 2.94 K1 25661 * S a g ltta rll 3.30 B8 25941 Nunki a S a g ltta r ll 2.14 B3 *26161 A scella C S a g ltta rll 2.71 A4 30942 A lnair a Gruls 2.16 B5 Second Flight Boss catalog Star name Bayer name V isual Spectral number magnitude, class °v 519 Ankaa a Phoenicia 2.44 G5 9188 Adara c Canis Majorls 1.63 B1 9443 Wezen 6 Canls Majorls 1.98 G3 9886 Aludra n Canls Majorls 2.43 B5 13926 Regulus a Leonls 1.34 B7 *14177 Algleba Y1 Leonls 2.61 K0 18643 Alkaid n Ursae Majorls 1.91 B3 25466 Vega a Lyrae .14 A1 30491 Deneb A lgledi 4 Caprlcoml 2.98 A5 32000 Fomalhaut a Plscis Austria! 1.29 A2 *Double star 138 name, constellation, visual magnitude, and spectral class. The lowest Intensity visual magnitude star Identified from the star mapper data of the first flight was a B8 star with a visual magnitude of +3.30 and the lowest Intensity star, relative to the Instrument spectral response, Identified from data of the first flight was an M3 star with a visual magnitude of +3.19. The results of the second flight were very similar and a listing of all stars which were detected and identified during the second flight are also shown in Table 13. It can therefore be seen that the design goal of detecting and identifying stars of a +3 visual magni tude class was met in both f lig h ts . The number of stars per spin period sighted by the star mapper during each data period varied from 3 to 14. Ambiguities in associating raw data with vertical or slanted slit transits and in star identifica tio n were encountered when two o r more s ta rs w ith sm all angular separa tions were present in the fleld-of-vlew of the star mapper. Such problems usually resulted in unusable data from one or both stars, in which case the questionable stars were eliminated from the star identi fication list. As a result, the number of stars actually used for attitude determination was considerably less than the number identified as shown in Table 14. A priori data were used in both the star and parameter identification program in the form of estimates of nominal vehicle trajectory, vehicle moments of inertia based on preflight measurements, and an approximation of vehicle spin rate based on suc cessive sightings of identified stars. A summary of these a priori data is given in Table 15. 139 TABLE 14 STARS USED IN PARAMETER ESTIMATION PROCEDURE Boss catalog Apparent S pectral nunfcer Name Bayer name visual class magnitude First Flight *3584 Acamar 0 Eridanl 3.06 A2 6274 Cursa 0 Eridanl 2.92 A3 6668 B e lla trix Y O rlonis 1.70 B2 6847 Mintaka 6 O rlonis 2.48 09 15145 Merak 0 Ursae Majorls 2.44 AO 16268 Phekda Y Ursae Majorls 2.54 AO 17518 A lioth c Ursae Majorls 1.68 AO 18643 Alkaid n Ursae Majorls 1.91 B3 20947 Alphecca a Coronae Borealis 2.31 AO 22193 Kornephoros 6 Herculls 2.81 G8 *26161 A scella C Saglttarll 2.71 A2 Second Flight 519 Ankaa a Phoenicia 2.44 G5 9188 Adara c Canls Majorls 1.63 B1 9443 Wezen 6 Canls Majorls 1.98 G3 9886 Aludra n Canls Majorls 2.43 B5 13926 Regulus a Leonls 1.34 B7 *14177 Algleba Y^ Leonls 2.61 K0 18643 Alkaid n Ursae Majorls 1.91 , B3 25466 Vega a Lyrae 0.14 Al 30491 Deneb Algledl 6 Caprlcornl 2.98 A5 32000 Fomalhaut a Piscls Austrlnl 1.29 A2 *Double 8tar 140 TABLE 15 A PRIORI DATA P lig h t 1 F light 2 Preflight Measurement Yv» deg 2.865857 2.861182 Yfl. deg -0.196679 -0.202650 8V, deg 0 0 V deg 43.151340 43.115963 Ij, slug-ft2 129.92 135.2 Ij, slug-ft2 8.27 8.756 Launch Data and Initial Flight Estimates w , deg/sec 306.3 239.6 68 C Ej. deg Unknown Unknown e2» deg Unknown Unknown 9, deg 75.42 137.5 0. deg 50.95 58.5 *4>» deg/sec 19.5 15.52 ♦0» deg Unknown Unknown deg/sec 286.8 224.08 ♦0 » deg Unknown Unknown 0, deg Unknown Unknown *4 was estimated with the assumption that cos0 » 1. 141 Parameter Estimation and Pointing Direction Computation Results Had the vehicle actually been a symmetrical, rigid, torque-free, spinning body, the vehicle's motion for an entire data period could have been represented by a single set of nine vehicle motion parameters. Under this assumption the single set of nine vehicle motion parameters could have been determined by performing a least-squares fitting of all the star transit times In any data period. However, finite errors In the estimates of certain parameters of X were shown to cause errors In pointing direction computations to propagate with time (see Appendix G). Hence, the maximum length of a data fitting period was selected to be approximately one precession period In duration. Further, the results of the simulation studies of Chapter 6 demonstrated that a lower limit for the data fitting period should be approximately 3/4 of a precession period In duration. This latter limit is the result of an assumed asym metry of 10 per cent in the moments of inertia 1^ and I^* In addition, preliminary evaluation of the results of the attitude determination pro cedure for both flights indicated that some undefined source produced small disturbing torques during the data periods. Presence of these torques was noted by examining parameter estimates and relating their values to those expected for the assumed dynamical model. A description of these torques and their effects on the vehicle dynamics is presented In a la te r se ctio n . However i t should be noted here that some data fitting periods were selected to range from approximately one-half to three-fourths of a precession period in order to reduce the effects of the uncertainties in the vehicle dynamics. For example, the second data period of the first flight was divided Into 12 time Intervals 142 and the 12 sets of vehicle notion parameters were estimated in order to define the vehicle's motion over the entire data period as shown in Table 16. In the simulation studies of Chapter 6 the convergence factor M was adjusted until a least-squares iteration resulted in time reBidual values At^ which agreed with the added time errors, c ^ , to within an accuracy of 5 microseconds. Zn these studies the standard deviation of At^ was calculated for each set of k stars used in a least-squares solution. This standard deviation of the time residuals, o , was found to be approximately 20 microseconds for all cases. For 0t r w 20 micro seconds, the accuracy of the computed pointing directions was within the specified requirements of 0.006° to 0.01°. This parameter, otr> was the factor which was used to assess the adequacy of a least-squares solu tion for flight data. In the case of flight data, the convergence factor M was adjusted u n til o t r was near i t s minimum value. When o was a t an acceptable minimum value, a least-squares solution was accepted as an adequate one. Individual values of At^ were also used to assess the adequacy of each least-squares solution using flight data. If one or more values of At^ in a set of k data points were inordinately larger than the re maining menfcers of the set, the questionable data points were redxamined. In every case the large values of At^ could be attributed to spurious errors in range time counting, incorrectly cataloged right ascension or declination values of the star associated with the particular data point, or false star identification resulting from two or more detect able stars appearing in the star mapper field of view simultaneously. TABLE 16 VEHICLE MOTION PARAMETERS FOR THE SECOND DATA PERIOD OF FLIGHT ONE Time 0, 0 O» 0 , *o* 0 . 0, e, (a) deg 61’ e2* deg deg/sec deg deg/sec deg deg deg deg min sec 24 3.23498 •0.315605 257.490643 288.445858 i 107.284283 18.482840 76.451450 54.130412 0.038607 0.046579 24 14.08154 -.312157 144.903893 287.062817 308.978760 j 19.882012 76.461610 54.137077 .039603 .045426 24 25.51978 .312637 6.962404 288.093067 357.829037118.865309 76.456949 54.137756 .027676 .043253 24 33.99485 -.304882 107.741646 287.259624 338.545620 19.710493 76.457376 54.132216 .033905 .044052 24 41.93850 .309047 48.131818 287.881664 316.643620 19.100499 76.454263 54.140100 .040073 .045732 24 52.84173 .310211 307.467949 287.377155 164.405958 19.618340 76.455278 54.144165 .038900 .047986 25 3.49465 .314057 < 129.948818 287.237286 12.317471 19.770194 76.463597 54.134274 -.000252 .042944 25 15.94554 .308249; >107.170229 287.945621 257.622326 19.075793 76.455045 54.128122 .004248 .041465 25 27.93654 .314447 317.269981 287.772449 129.021713 19.257796 76.454346 54.140325 .023696 .042999 25 36.32649 .305107 158.666159 287.572033 287.587963 19.466666 76.453516 54.141089 -.002766 .046548 25 46.69653 .310320 315.331436 287.206608 130.950977 19.842483 76.438763 54.144919 .045223 .049947 25 57.27342 .314742 114.694677 287.473938 339.198837 19.586070 76.445923 54.141324 .029057 .047476 aReal time after 0600 UT, August 16, 1966. 144 The errors in range time were relatively easy to locate since a taped tlme-hl8tory record of range time counting was available. Flight data results revealed that several stars consistently exhi bited large time residuals. Subsequent checking traced the cause of these high residuals to errors in the updated star positions based on values listed in the Boss catalog. The positions of these stars (6 Orlonis, n Ursae Majorls, and a Piscis Austrlni) were cross-checked with positions taken from the Apparent Place Tables [ref. 33] and updated to the nearest 0.1 day of the flight. Discrepancies of the order of 0.015° in the star positions taken from the B o b s catalog and updated to the nearest 0.1 day of flight were found. When the star's positions deter mined from the Apparent Place Tables were used for these stars, the time resldualB dropped to acceptable lim its. To illustrate the use of the nine vehlcle-motion parameters, a set of these parameters was selected from the second data period of the first flight and used to generate time histories of both the star mapper opti- A ^ cal axis lg and the vehicle principal spin axis kg. The results of these computations are shown in Figures 25 and 26 in which the pointing directions of these two axes are plotted in terms of declination and right ascension for selected time periods. Figure 25 represents the pointing direction of the optical axis during a single spin period and Figure 26 represents the pointing direction of the vehicle principal spin axis during a complete precession cycle. Figure 27 represents the A pointing direction of the kg axis during the same precession cycle and illustrates the effects of the misallnement angles, Cj. and Cj* The complete listing of all the seta of nine parameters for each data period and for both flights was used in a similar manner to generate time Dtcllnatlon, Figure 25. Pointing Direction of the Star Mapper Optical Axis (ig) for One Vehicle Spin Period Spin Vehicle One for (ig) Axis Optical Mapper Star the of Direction Pointing 25. Figure •20 -90 -60 -30 T - -TO 60 -C 0 2 80 O - TO - to 20 20 to 0 0 0 O O lO CO 0 9 60 70 0 6 30 0 4 0 3 Period of P ro je c t Scanner F lig h t 1 fo r Time Beginning 6h 25m 46.7 s UT on on UT s 46.7 25m 6h Beginning Time r fo 1 t h lig F Data Scanner Second t the c from je ro Selected P of Parameters Motion Vehicle Period Nine of Set the on Based uut 6 1966. 16, August 120 • • O O o n MO OO i r i RigM otc*ision, <5«j RigMotc*ision, 160 T (0 9 200 30 220 20 « 20 60 20 9 20 0 3 0 3 3 3 3 0 36 130 0 34 330 20 3 3C 300 290 290 270 0 26 230 z«0 230 0 2 2 0 3 0 0 2 r90 (80 ITO . "* . • • 1 • w • • • ITU o w n u t 145 1 iue 6 Pitn Dieto o Vhce i pal pn xs g or n Peeso Cce Based Cycle Precession One r fo kg Axis Spin l a ip c rin P Vehicle of irection D Pointing 26. Figure Declination, deg 36.25 35.65 35.85 36.05 n h Se o Nn Veil Mto Prmtr Slce fo te eod aa Period Data Second the from Selected Parameters Motion ehicle V Nine of et S the on f ojc Sanr Flght f Tm Bgnig i 2m 67 U o Ags 1, 96 ££ 1966. 16, August UT on 46.7s 25m fib Beginning Time r fo 1 t h lig F Scanner ject ro P of gt seso, deg ascension, ight R s a 3625 r STARTING POINT ( 6h 25m 46.7s UT ) 36.05 DECLINATION, > ENDING POINT deg (6h 26m 5s UT ) 3 5 © + 35.65 35.45 346.05 346.25 346.45 346.63 346.85 347.05 RIGHT ASCENSION, deg Figure 27. Pointing Direction of kg Axis for One Precession Cycle Based on the Set of Nine Vehicle Motion Paraaeters Selected froa the Second Data Period of Project Scanner Flight 1 for Tine Beginning 6h 25a 46.7s UT on August 16. 1966. 148 A histories for the ig axis. An analysis of all solutions of the nine veh id e-motion parameters determined from flight data showed a o value of approximately 30 micro seconds. In the simulation study described In Chapter 6, star transit time errors having a standard deviation of approximately 21 microseconds were added to the simulated star transit data. These time errors result ed in a o^ value of approximately 20 microseconds when data were fitted over an interval of 0.5 to 1.50 precession cycles. In Chapter 6, the maximum standard deviation (o^) of the pointing direction error of the A * ig axis wa8 determined to be in the range of 0.003 to 0.004 degree for at least three or four stars per vehicle spin and a lower limit of data fitting period of approximately 3/4 a precession period. Results of Chapter 6 and Appendix 6 show that and o^ are proportional to e and o respectively. Since it has been shown that o and o tv c cr have very close agreement, it can be said that there is approximately a proportionality relationship between o^ and o ^. Therefore, the standard deviation in pointing direction error determined from flight data is estimated to lie in the range of 0.0045° to 0.006°. Although the time residuals of the flight data were somewhat larger than the transit time errors used in the simulation study, the estimated accuracy of the flight data still remained within acceptable limits for the flight experiment. It should be noted that the value of obtained from flight data is approximately 1.5 times the value of o obtained from simulation data. This factor of increase in measurement noise is in close agreement with the 1.3 factor of increase in rms noise noted in the earlier discussion on the electrical signal and noise characteristics observed from flight data. 149 Relationship of Data Anomalies and Dynamical Model Inaccuracies Errors In the estimates may arise from sources other than observa tion errors. The primary source of these additional errors is inaccuracies in the assumed model. For the dynamical system of this study, errors in the assumed dynamical model were considered to be deterministic in nature rather than stochastic and certainly not of a white noise nature. These types of errors were further considered to be of the type which would cause bias errors which would manifest themselves in anomalies in the parameter estimates, residuals of the parameter estimation results, and basic dynamic behavior of the assumed model. This reasoning led to a decision to use a weighted leaBt squares estimator rather than a Kalman estimator. Further, the parameter estimation procedure was treated as an initial value problem with solutions obtained for sequential time l periods with duration of 1/2 to 3/4 of a precession period. This prob lem formulation and discrete data fitting period definition led to the decision to use a batch processing estimation technique rather than a sequential estimation procedure. This section represents a brief dis cussion of two anomalies noted in results of the parameter estimation procedure and describes the relationship of these anomalies to errors or omissions in the assumed dynamical model. Periodicities in the Estimation Residuals If the measurement or transit time errors were truly random, the time residuals of the estimation procedure would exhibit similar random characteristics. In the case of flight data, a pronounced periodicity was observed in the time residuals when these residuals were plotted as a function of time. The period of the cyclical variations in the time residuals was found to have a value of approximately 3.2 seconds. Since this periodicity was found to occur throughout all data periods of both flights, some biasing effect.due to an error in the assumed space craft dynamical model was suspected. Both the payload and X-248 rocket motor were reexamined to determine if there were any contributing dynamics from these sources. A possible source of additional dynamical terms was located in the experimental package of the payload. As noted earlier, this package contained two infrared radiometers used to experi mentally measure characteristics of the earth's atmosphere near its horizon. These radiometers contained two mirrors which were scanned A A nominally in the ig - kg plane. This scanning action was found to cause a variation in the moment of inertia about the principal spin axis since -3 2 -3 the scan caused 1^ to vary + 1.02 x 10 slug-ft and -1.40 x 10 slug- 2 2 ft about its nominal value of 8.3 slug-ft or +0.0122 and -0.0172 res pectively. For the case of the assumed dynamical model the amplitude of the angular momentum vector, h, is a constant. The definition of h from Chapter 2 Is *3 W3 h ■ *t • <2-7> For h and 6 to remain constants, we have ( I 3 + AI3) ( w3 ± Au 3) - K *(7.6) where K is a constant. Then the change in angular velocity is re la te d to a change in the moment of inertia I3 by -3 2 -3 2 For the AI^ values of +1.02 x 10 slug-ft and -1.40 x 10 slug-ft , the corresponding values of Aco^ are -0.037 deg/sec and +0.052 deg/sec respectively. It was further determined that these variations In I^ and u>2 varied with a frequency equal to that of the frequency of the scanning mirrors and with a period of 3.2 seconds. These variations In the moments of Inertia and resulting variation In angular velocity were considered to be negligible. It may also be noted that the maximum predicted change in the moments of inertia based on the mirror scan motion was 0.017 per cent. Simulation studies indicated that asymmetries of this amount would produce negligible error in attitude determination when a symmetrical model was assumed. The important point is that the parameter estimation procedure was sufficiently sensitive so that time residuals of the flight data produced periodicities which were directly attributable to ignored but known sources which contributed to the vehicle dynamics. Non-Zero Angular Acceleration and Non-Zero Control Torques Preliminary analysis of results of the attitude determination pro cedure for both flights indicated that some undefined source produced small disturbing torques during the data periods. The torques manifested themselves by producing non-zero angular accelerations. For a symmetri- cal, rigid, torque-free body the angular velocity has been shown to have a constant value. However, flight results indicated'that this parameter continued to increase In magnitude throughout each data period and throughout each flight. The discrepancy between the model dynamics assumed for the motion of the vehicle in the parameter-identlficatlon program and the apparent 152 flight vehicle dynamics Is seen by comparing the observed spin rate and the predicted spin rate which Is defined by r|> + $ cos 9. Values of determined from sequential solutions of the parameter estimation proce dure were found to be slowly but constantly increasing. This constantly Increasing spin rate was characteristic of all four data periods of both flights. The spin rate values obtained from the second data period of the first flight (see Table 16) are shown in Figure 28. The Increase of over the time period shown In this figure was 0.0A per cent. This percentage of increase for was approximately the same for all data periods of both flights* Since of Figure 28 appears to be a constant, it was considered highly probable that the disturbing torque was also a constant. A possible source of thiB disturbing torque could have been incomplete closure of the control values of the cold gas reaction jet control during the star mapper data gathering periods. An examination of the design data of the cold gas reaction jet control system, discussed earlier in this chapter and illustrated in Figure 19, revealed the specifications listed in Table 17. The units, SCF/mln, refer to standard cubic feet per minute of gas at standard temperature and pressure where the standard temperature and pressure are defined as 0°C and 14.7 psi, respectively [ref. 34]. Using this 3 definition, a standard cubic foot of nitrogen is 0.078 lb/ft [rerf. 35]. The Initial and final values of pressure are related to the total control activation times of Table 12. The specifications of Table 17 which could contribute to an acceleration torque are the jet mlsalinement and leakage tolerances. The effect of a jet mlsalinement is illustrated in Figure 29. The thrust developed by emission of the gas may be 307.10 Increasing Spin Race u 9 Assumed Constant Spin Race 307.05 -a9 9 <3 307.00 5a. m H 9 9 U M O. U H« a o 306.95 u« o H 306.90 375 400350 425 450 475 Elapsed Time in Flight, sec Figure 28. Increase of Total Angular Spin Rate During Che Second Data Period of Scanner Flight 1. m u 154 Spacecraft Shell Nominal J e t Allnement M isalined J e t Total Thrust Thruat due to Mlsallneaent Thrust which Produces a Precession Figure 29. Illustration of the Source of an Acceleration Torque due to a Misalined Jet. 155 TABLE 17 CONTROL SYSTEM SPECIFICATIONS Fuel - nitrogen Resevoir volume - 200 in 3 Initial pressure - 3000 psl Final pressure - 100 psl Maximum th ru st - 5 lbs + 0 .5 lb (a t 3000 p sl) Allowable leakage - 0.0005 SCF/mln Allowable mlsalinement of thrust vector - 2.2° expressed as T - q(t)v(t) + p(t) Ae (7.8) where q(t) is the mass flow rate m(t), v(t) is the velocity of the emitted gas* p(t) Is the exit pressure of the gas* and A is the effective area of the exit orifice. The value of q(t) will be assumed to be a constant value equal to the maximum leakage of Table 17. The quantity v(t) may be determined by use of the ideal gas rela tio n sh ip [ r e f . 36] p(t)V ■ Mv^ .(7.9) where p(t) - pressure V ■ container volume M » gaa molecular weight v(t) - velocity of gas 156 The value of A Is assumed to be unknown for the case of deactivated e controls or for a closed valve condition. It Is the intention of this section to demonstrate that the angular acceleration arising from the « quantity qv is of the same order of magnitude as the value of noted In the flight data. For a molecular weight of 28 grams per mole [ref. 35] and the values of p and V from Table 17, the gas velocity, v, has a value of approximately 4830 feet per second. For simplicity the velocity is also assumed to be constant. The resulting constant thrust due to qv -4 of equation 7.8 is approximately 10 lb. The tangential thrust due to a maximum jet mlsalinement of 2.2° is 3.84 x 10 ® lb. From Table 10 the spacecraft radius is seen to be 1 foot. The resulting constant torque, L, about the kg* axis is then 3.84 x 10 ft-6 lb. The angular acceleration, Ziy due to an assumed constant torque about the kg axis may be found from equation 2.23 to be u j * a3 ’ <7-10> For the 1^ value of 8.3 slug-ft and the above value of L, is found -4.2 to be approximately 0.3 x 10 deg/sec . The value of noted from the flight results plotted in Figure 28 is seen to be approximately 10 x 10 ^ deg/sec2. These two values agree sufficiently closely In magnl- * tude for a gas leakage condition during the deactivated control periods to be considered a possible source of the constant torque giving rise to the angular acceleration. It should be noted that the leakage speci fication of Table 17 was not verified and could have been greater than the listed value. In addition, the value of A wase unknown and further was assumed to be zero in the calculations. However, it is certain that 157 A could not be zero for any finite amount of gas leakage. Increases 6 In q and a non-zero value for Aq could Increase the computed value of thrust and cause the computed value of to agree more closely with the value obtained from flight data. It should be noted that the of the flight data produced an approximate change of 0.04 percent In the nominal value of over a time period of 100 seconds. The Important point to be made here Is that the parameter estimation procedure was sensitive enough to detect errors In the assumed dynamical model which produced extremely small deviations In the assumed vehicle motion. Summary of Flight Results The discussion of flight data In this chapter covered several factors in addition to results of the parameter estimation procedures which were used for spacecraft attitude determination computations. A summary of all results noted in this chapter Is given in Table 18. The significant factors of Table 18 are: (1) the star signal and noise background characteristics* and anomalies noted In the parameter esti mation results that pointed out errors In the catalog positions of three stars detected during the flights and Inaccuracies in the assumed dynamics of the spacecraft; (2) estimates of errors in the transit time data; and (3) estimated accuracy of pointing direction computations for t the ig axis determined by UBe of results of the parameter estimation procedure. These results indicate that the goal of detecting stars of +3 mv and brighter was met and exceeded. Identification of these detected stars provided at least 3 stars per vehicle rotation for appli cation to the parameter estimation procedure. The noise levels en countered in the flight data and results of the parameter estimation 158 TABLE 18 SUMMARY OF FLIGHT RESULTS 1. Total time of data (approximately) 10 minutes 2. Data fitting period time intervals 9-18 seconds 3. Detected stars per vehicle spin period 3-1 4 4. Weakest star detected (visual spectrum) +3.3 m B8 v 5. Increase in star signal sensitivity 1.5> predicted 6. Increase in rros background noise 1.5> predicted 7. Estimated noise in transit time measurements 30 usee, (la) 8. Estimated accuracy in attitude determ ination of s ta r mapper 0.006° (la) optical axis 9. Large time residuals related to 6 Orlonis catalog star position errors n Ursae Majorls (approx* 0.015°) a Piscis Austrini 10. Dynamical model Inaccuracies (a) Residual periodi detected and related to probable cities of 3.2 sec sources period (b) Linearly increas ing approx. 0.04JS/2-1/2 mln. procedures Indicated that the accuracy goal was met for attitude deter mination of the star mapper optical axis. Anomalies noted in the re sults of the parameter estimation procedures were detected and related to either proven or probable sources. CHAPTER 8 RESULTS, CONCLUSIONS AND RECOMMENDATIONS As stated in the introduction, the general objective of this dissertation Is to develop a technique suitable for attitude deter mination of a spinning spacecraft through the application of estimation procedures, given the dynamical model of the spacecraft, to the times of sightings of stars as observed in a sensor reference frame attached to the spacecraft. In this concept, spacecraft spin motion scans the field of view of a passive optical system about the celestial sphere. In this development an effort has been made to achieve high levels of both accuracy and reliability through the use of a relatively simple measurement technique with sophistication of elaborate Instrumentation being replaced by an Instrument containing no moving p a rts and a moder ately sophisticated data processing technique. Specific objectives include development of a technique for detection of observed stars, a method for identification of detected stars, and a procedure for esti mation of the parameters defining the dynamical equations of motion and spacecraft and sensor geometry through use of measured times of sightings of the detected and identified stars. Estimation of these parameters provides the necessary quantities for determination of the attitude of selected axes of those reference frames defined to be attached to the spacecraft. Results The dissertation covers consideration of the design aspects of the 160 161 star mapping Instrument, including a discussion of advantages of the application of optical coding techniques to the detection of star signals. The goal in this part of the study Is the enhancement of dis crimination of stars brighter than a limiting visual magnitude against the background of dimmer stars. A technique of identifying the detected stars is presented. Procedures for determining the attitude time history of selected spacecraft axes using the sighting times of the identified stars were developed. These procedures involve the application of a nonlinear regression analysis method to the determination of unknown, or poorly known, parameters in the spacecraft dynamical equations of motion and problem geometry. The accuracy of this attitude determination procedure was investi gated through use of computer simulation studies. In these studies an exact spacecraft dynamical model and statistically independent star sighting time errors were assumed. A Monte Carlo study was conducted to determine the sensitivity of the attitude determination procedure to such variables as sighting time errors, star placement, and nunfcer of sighted stars. This study produced a prediction of the expected accuracy of the attitude determination procedure. Simulation studies were also used to validate the feasibility of establishing criteria to be used as measures of "goodness" of estimates of the dynamical equation parameters in a real world situation. The presented attitude determination procedure is illustrated through application of the estimation technique to flight data obtained from two sub-orbital probes launched from Wallops Island, Virginia. The flight vehicles used for these launchings may be very closely approxi mated by symmetrical, torque free, rigid bodies for the flight period 162 of Interest. Star napping was accomplished by use of a passive optical instru ment consisting of a telescope, a coded reticle, and a photomultiplier. The telescope was mounted w ith i t s o p tic a l ax is normal to the spacecraft principal spin axis. Spin motion about this axis caused star images to transit the reticle with each star transit sensed by the photomultiplier as two sequences of coded pulses that were telemetered to a ground receiving station. The amplitudes of the received pulses were propor tional to the photometric magnitude of the stars, and the time separation of two pulse groups generated by a single star transit was related to the elevation angle of the star within the field of view of the tele scope. Time separations of pulse groups generated by different stars were related to their azimuth angle separations. It was possible to correlate flight measurements to laboratory calibration. The results demonstrated that the goal of detecting and identifying stars in a +3 visual magnitude AO spectral class was achieved, the photometric calibration was verified within a factor of 2, the system fidelity provided star transit time measurements within the required resolution and accuracy, and the theoretical predictions of signal-to-nolse ratios were in close agreement with flight measure ments. A significant problem that was not resolved was the design and construction of a sun shield having the required attenuation of stray sui.light to allow daylight operation. Identification of the sensed stars was made on the basis of their angular separations in the celestial sphere as determined from their transit times. Information consisting of the cataloged positions of 163 these Identified stars and their measured transit times was fitted In a least-squares method to a set of equations representing an assumed dynamical model of the spacecraft attitude. The dynamical model was developed on the assumptions that the vehicle was spin-stabilized, symmetrical, and free of external torques. The least-squares fitting of star position and transit time data resulted in definition of sets of parameters of the assumed dynamical model. The equations of the model and the sets of parameters were then used to generate time histo ries of selected reference frames associated with the model. In partic ular, time histories were determined for pointing directions of the vehicle principal spin axis and the star mapper optical axis. Accuracy of the solution of the attitude determination problem was based on the pointing direction accuracy of the star mapper optical axis. Results of these flights demonstrate that the weakest signal de tected was produced by a +3.3mvB8 star, both signal sensitivity and back ground noise were increased by a factor of approximately 1.5, and a tti tude determination was achieved with an estimated accuracy of 0.006°. The estimator procedure was sufficiently sensitive to detect errors in catalog positions of several stars of the order of 0.015°. Effects of inaccuracies in the assumed spacecraft model on the attitude determina tion resultB were investigated. These Inaccuracies Include asymmetries, periodically varying moments of inertia, and constant torques. The results demonstrate the feasibility of obtaining extremely accurate estimates of attitude determination through the use of relatively simple instrumentation coupled with detection, identification and estimation techniques such as presented herein. Operational systems for long term 164 missions will requite refinements such as additional observables (I.e., disturbance toruqes and a sun sensor for continuous daylight operation) and compensation for systematic errors In the assumed model due either to negligent omission or to long term dynamical changes. Conclusions and Recommendations The results of this dissertation lay the groundwork for more exten sive research In the area of attitude determination of spinning space craft or for three axis stabilized vehicles employing scanning optical systems, and possibly for the solution of the non-linear problem, in cluding asymmetries and both control torques and external disturbance torques. A comprehensive Investigation has been made of this problem for the case of a nearly symmetrical spinning body containing a passively scanned star sensing Instrument. For application of the reported attitude determination technique to long duration space flights, several areas of the data-processing pro cedures should be modified to facilitate data handling. The extent of the modifications depends on the particular application. However, In any future application It would be desirable to eliminate manual data handling as much as possible, particularly In the areas of pairing apparent star observation times and star identification. Increased * automation In these areas may require that the entire data-processlng procedure be a closed-loop operation. For application to an orbiting spacecraft, a more exact model of the dynamical equations of motion would be required. This model should include the effects of vehicle asymmetries and at least the moBt significant external disturbance torques for the spacecraft under consideration. Methods other than the 165 reported least-squares method used for the solution of the undetermined parameters In the equations of motion should be Investigated. In addition, sequential updating methods such as the Kalman filter should be studied for application to long-life orbiting spacecraft. In particular, the star Identification procedure described in this dissertation requires that the direction of the angular momentum vector be known to within +10°. This requirement is associated with the simul taneous star identification and accurate pointing direction determination requirement. A key to successful autonomous Inertial attitude determi nation lies In the separation of the task of determining the accurate direction to a star from the task of star Identification. This separa tion could result in Identification of detected stars without the need for a priori information regarding initial vehicle orientation in celestial coordinates. Two approaches to the identification problem for a randomly oriented spacecraft are described by Gorsteln [ref. 37]. The first approach uses digital computation techniques to achieve auto matic star pattern recognition for all orientation situations. The second approach employs holographic storage of reference maps of cata loged stars and an application of matched filter theory to an optical star field correlation system. The second of these approaches Is considered to be the most promising for the immediate future from an implementation point of view. Development of a system.employing holo graphic storage has progressed to the point where candidate designs such as that described by Welch [ref. 38] have been proposed. When the star data processing scheme of this dissertation was evolved, the importance of sequential processing was minimal because each data period could be chosen for short intervals of time and a 166 closed £orm solution for the assumed mathematical model of the space craft dynamics was available making updating easy. However, for orbital attitude determination problems, data extending over many orbits are required, and as a result of long term effects of small torques, the closed form solution of this study Is inadequate to handle the problem over these extended periods. However, It Is recognized that development of recursive procedures might result In significant analysis and data processing advantages. In general the solution procedures developed by Cox [ref. 39] and Friedland [ref. AO] should be directly applicable to the attitude determination problem since both the measurements and the differential equations in their developments are nonlinear. There is an additional problem which should be considered. No matter how many terms are employed in the mathematical model of the I physical system, there are always going to be effects inaccurately or incorrectly described. Generally, the Kalman procedures represent these Inaccuracies as stochastic variables, usually uncorrelated and GausBlan, but always with zero mean. Physically, this means the assumed disturbance effects are noises which may change rapidly, but on a long time average, these effects are zero. It is desirable to show the effect of an unknown, non-random disturbance on the convergence of the system, tje ., w ill the estimation procedure maintain its accuracy or will it slowly drift from the true state until at some point, it will require an extensive correction or reinitialization. The Kalman filter has been used in a number of nonlinear aerospace applications. However, when the disturbing noise or system uncertainty, v» is small or does not have a character which can be readily approximated by white noise, the filter 167 has given inaccurate results* This is mainly due to the fact that terms of the convariance matrix, P, become so small that the filter becomes Insensitive to small errors* Several concepts have been developed for alleviating this Insensitivity problem. Among these concepts are the limited memory filters developed by Schmidt [ref* 41] and Jazwlnski [ref. 42]. Another problem related to sequential pro cessing using the recursive Kalman estimator is that of the effect of each subsequent measurement on the basic observability of the system. A solution to this problem, proposed by Nayak [ref. 43], is to make available for each measurement a particular vector suitable for use in judging whether the measurement has Increased or otherwise modified the observability of the system. The approach further provides a means for conditioning of the measurement and system estimator and also provides a basis for a decision criterion. Lastly, measurement noise should be considered to be dependent upon the magnitude of the detected star and should enter the problem with appropriate weighting. APPENDIX A DYNAMICAL EQUATIONS OF MOTION FOR Alt ARBITRARY SPACECRAFT For the purpose of deriving the equations of motion of the spacecraft about its mass center, the rate of change of the magnitude of the angular momentum vector of the system w ill be considered in addition to the previously defined Euler angles 4>, 0, 8 and ij/. In addition the body w ill be assumed to be subjected to control torques. The development here w ill result in the common dynamical system state equations used to describe the motion of an artificial earth satellite [ref. 44]. According to the basic equations of dynamics, the time derivative of the angular momentum of a rigid body is equal to the torque due to forces acting on the body. From classical mechanics, this relationship is [re f. 45] w*iere is the angular momentum of the body and is the instantaneous torque. The subscript (1) on P and L denotes the reference frame for which equation A.l is valid since this relationship holds only in an AAA inertial reference frame. In an earlier discussion, the (i^, j^, k^) reference frame waB assumed to have a fixed orientation, defined by $ AAA and Q, relative to the (i^, k^) reference frame. In this 168 . , lfi9 development, the angles 4 and 0 w ill be assumed to have non-zero time • * derivatives. For non-zero 4 and 0 the derivative of £ may be expressed as dP, dP -JT * -3T + S-Pj • <*■» It Is to be no^ed that equation A.2 Is merely symbolic, and to obtain the components of - would require a formal transformation of the right- hand side of this equation. The quantity n. is the angular velocity due • • * * * to 4> and 0 and is related to the (1 ^* J ^ » ^3 ) reference frame by n - i kL + 0 i3 . (a.3) Using the relationship (A. 4) we have fi - i sin 0 jj* ® i^ + * cos 0 k^ • (A.5) 0 k For convenience it is assumed here that Is directed along the axis with the magnitude of P^ denoted by h. Then the torque vector expressed in the (i^* $3 * k^) reference frame Ib Now using the relationship for the total angular velocity of the system we have flj, - * + O i3 + * k3 + + * kfi . (A.7) The angular rates of equation A.7 may be expressed in terms of angular rates in the body system or the (ig, jgi kg) reference frame through use of the transformation of equations A. 5, A.8, and A.9 0* A * _ a • ‘ 3 *6 al l a12 a13 *6 A ■1 (A. 8) J 3 - <40(0)010 h a21 a22 °23 ^6 A U3. .kg- -a31 a32 a33 V V * • (A. 9) u - <0)010 ^6 A Lv .feg. The matrix elements a^ of equation A.8 are defined in Chapter 3. The body rates are denoted here as and for the components in the ig, jg, and kg directions respectively. These components of angular velocity are then 171 ■ *(cos 0 a ^ + aln 0 a ^ ) + 0 a^ + a^ ♦ + cos ^»0 (A. 10a) (i)2 “ . 4(cos 0 a^j + sin 0 a,,2) + © a^2 + a32 ^ " Bin (A. 10b) * * • i • 4(cob 0 + sin 0 + ® + a 3 3 ♦ + ♦ * (A* 10c) On the other hand* the components of the angular momentum vector In the body reference frame can be expressed as p , ■ Iii) (A. 11) —O “ or ■?i 6' ‘W - ?J« h “z P L k6J - V v Equation A.12 is, of course, valid only for 1^, 12, and Ij being the p rin cip al moments of in e rtia . The angular momentum vector Pg may also be expressed as a formal transformation of P^ as P3 - (♦XeX*)P< (A. 13) or 0 0 (A. 14) Lh, 172 Then 1 I P* ___ AO *h sin 0 sin - h sin 8 cop f (A.15) PJ6 _h cos 6 • Pk6- From equations A.12 and A.15 we have "(h/1^) sin 8 sin ■“l ' (h/l^) sin 9 cos ^ (A. 16) “2 - ,(h/I3) cos a -w3- and from equation A.6 0 ■ -L23/h 4 - l*i3/h sin 0 (A. 17) Substitution of equations A.16 and A.17 into equation A.10, and solving for 6, and ^ results in sin2 « cos2 \|i \ / \ (- j — + - j - - J - -X ” ^L13 cos* + LJ3 sin♦ j -L 13 cot 0 (A.IB) 173 sin 9 cos 4* sin if* (A ' A) i ( lj3 cos $ - sin ♦) (A.19) 2 2 sin i|i cos \ cos e h h * r ) . ^13 sin ft . ^*13 cos ft (A.20) h sin 0 h sin 9 Equations A,17, A.18, A.19, and A.20 are the dynamical equations necessary for describing motion and attitude of the vehicle for arbitrary torques and unequal principal moments of inertia. The three cases to be considered here are: zero torques and equal values for 1^ and Iji zero torques and slight asymmetry for 1^ and 1^• equal values for 1^ and Ij} and a small constant torque. Case 1.—Zero Body TorqueB—Symmetrical Body (I^ ■ I2 “ I) For this case the dynamical equations of motion become __ • _ e 0 • 0 ♦ h 0 - (A. 21) « ♦ h /l • 0 • 1/ i- A 31 JL -H i i J C08e- Equations A.17, A.18, A.19, and A.20 are the dynamical equations necessary for describing motion and attitude of the vehicle for arbitrary torques and unequal p rin c ip a l moments of I n e rtia . The three cases to be considered here are: zero torques and equal values for and > zero torques and slight asymmetry for 1^ and Ij! equal values for 1^ and Ij, and a small constant torque. Case 1.—Zero Body Torques—Symmetrical Body (1^ ■ I 2 ■ I) For this case the dynamical equations of motion become 174 • • From equation A. 21 It Is clear that ♦ and ij> are constants since h and 9 are zero. The expression for h In terras of the system dynamics Is yet to be determined. The components of torque In the body reference frame may be expressed as dPg h - -dT + H -J* (A.22) with components Li6 ' V l - “2“3(I2-V (A. 23a) (A.23b) ^ - Ijij - V 2 (Il"I2) (A.23c) or foe the symmetrical case with zero body torques 0 ■ 0 - X(“2+»>3«1) - IjU ji^ (A.24) l. 0 - I 3W3 . From equation A.24 it Is clear that both and Wg are characterized by simple harmonic motion and a constant: a>^ « a sin (Wt + p) ■ a cob (Wt + p) Wg “ a constant where (I - U) W . — _1 Wl( 0) p - ta n S ^ to jT M From equations A.10 and A.21 we also have ■ $ sin 6 sin i|> ti>2 ■ ♦ sin 0 cos ip a>3 - • • Solution for $ and (I - I 3> ^ - w ■ ------w3 ; l3 W3 ♦ r_ I cos 6 176 The angular momentum vector P^ can be expressed In terms of the body dynamics by making use of equations A. 12 and A. 13 13 - - I j $ sin 0 cos 0 sin ♦ + I 3($ cos 0 + - 1^ ♦ sin 0 cos 0 cos 6 -I3(i cos 0 + $t)cos $ sin 0 >(A.28) P^ ■ i sin 0 + cos 0 + <30cos 0 Substitution of equation A.27 into A.28 results in P13 - 0 P,, - 0 (A. 29) 33 P - ^ 3 k3 cos 0 Since we have previously defined the magnitude of Pj to be h then we have (A. 30) cos 6 This completes the description of the dynamical equations of motion expressed by equation A.21. Case 2.--Zero Body Torques and Small Asymmetry As discussed in Whittaker [ref. 46]» the solutions to equations A.18, A.19, and A.20, for this case, may be written in terms of elliptic functions. However, this general solution is inconvenient for numerical 177 computations and physical Interpretation. A more convenient solution form v lll be sought here. I f the body has two equal moments of in e r tia , the modulus of the elliptic functions, k, becomes zero and hence these functions degenerate Into the circular functions of Case 1. If two moments are nearly equal, k Is generally small in comparison to unity. For this small k Bateman [ref. 47] has shown that the elliptic functions can be written as a series of circular functions. The purpose here is to develop such a series. We w ill rely on the general solution as presented by Whittaker and will generally retain his notation. Let the th ree moments of In e rtia be given by I1 - I(l-e) I 2 - I(l+ e ) (A.31) *3 “ I 3* *1 - *2 > X3 or *1 - l 2 < X3 where -1 < e < 1 . Now e > 0 i f I 3 > I2 and e <_ 0 if Ij < I2> The orientation of the AAA principal axes of the body with respect to the (1^, j^» k^) reference frame can s till be specified by the three Euler angles 9, $, and 4). Using Whittaker's results [ref. 46] we may write cog e » (ainh V - g2 slnh 3y + q6 sinh 5y + ...) 178 (cosh y + q2 cosh 3y + cosh 5 y + * • ■) x (1 + 2q cos 2pt + 2q4 cos Ayt + ...) (A.32a) (1 - 2q cos 2yt -f 2q^ cos Ayt + ...) tan (Jj « (1 + 2g cosh 2y -f 2q** cosh Ay -f . . . ) (1 - 2q cosh 2y + 2q^ cosh Ay + . . . ) x (sin yt - q2 sin 3yt + g6 sin 5yt ...) (A.32b) (cos ut + q2 cos 3yt + q cos 5yt ...) » + + /jL + q slnh 2y - 2g4 sinh Ay+... ____ \ t 0 ' 1 1 - 2q cosh 2y + 2q^ cosh Ay - ...? tan- l / 2g s in 2yt slnh 2y - 2q^ sin Ayt slnh Ay t ... ) i ^ (A.: ^c) \1 - 2q cos 2yt cosh 2y + 2q^ cos Ayt cosh AyAy + . . . / The quantities q» y, and y» which will be defined in the subsequent derivation* nay be regarded as the constants which specify the motion 2 2 where either 1^ >_ > I 3 * IjC - h > 0 or 1^ <_ < Ij, IjC - h < 0 and q is the parameter of the theta-functions (related to the original elliptic functions) and 1b small if the modulus k is small and c is the total system rotational energy. From page 1A6 of Whittaker [ref. A 6 ) fc2 - where 179 8 » —*-C' + E > 0 i f E < 0 I c-h2 8 In order to make further approximations in equation A.33 we must be assured that re and se are both between <-1 and 1. This smallness can be expressed as er - 0(n) es ■ 0 (n) where terms of order n are between -1 and 1. Approximations may now be made in equation A.33 with the result that k2 - -2e(r + s)(1 + e(r + s) + e2(r2 + s2 +rs)) + 0(nA) . (A.34) From page 147 of Whittaker [ref. 46] we have So (A.35) 180 Mow the parameter y oust be determined. Again Whittaker [ref. 46] giveB the re s u lt 1 + 2q cosh 2y 4- 2q coah 4 y a 1 - 2q cosh 2y + 2q cosh 4y .. 2 . . itsil |_ \ 1 “ E / 1+es 1-erJ e e2 2 1 + I (r-s+2) + ~ (r-s+2) + 0(n3) . Hence q cosh 2y “ 0(n), and q cosh 2y ■ “g* (r-s+2) . (A. 36) Since q cosh 2y ■ 0(n), approximations can now be made in equation A.32 with the result n 2 2 cos 6 - (tanh Y ) Ll + 4q cos 2yt + 4q (2 cos 2yt - cosh 2y) - 4q3(cos 2yt)(coah 3y)J + 0 (n S (A.37) 2 tan(||»-^o) - (tan yt) [l + y (r-s+2) + ((r-s+2)2 - 1/2(r+s)2) + 0(n3)] (A.38) 181 ♦ ■ + 4y q(slnh 2y)(l + 2q cosh 2y + 4q2 cosh2 2y) + 0(n^)J t - 2q(sin 2|it) (slnh 2y)(l + 2q cosh 2y 2 x cos 2pt + 4q2 (cosh2 2y) (cos2 2yt) + x (slnh2 2y)(sln2 2pt)) + 0(nS . (A.39) In equation A.38 we may take the arc tangent of each side* and in equation A.37 we may set tanh y = cos 6 o I and then take the arc cosine of eadh side. Hence, 6 " ®o ” ^q(cot 0Q) [cos 2pt + q((2csc2 0 q )( cos 2 2yt) - cosh 2y)] + 0(n3) (A.40) ^ - *c + Wt + e^Bl" 2-H.O. [(r-s+2) + ^ ((r-s+2)2 cos 2pt - + 0(n3) . (A.41) How, equations A.39, A.40, and A.41 are not useful as given since they Involve the dependent parameters, 0O, r, and s. We choose to eliminate s from the system in favor of the more physically meaningful parameters r and 6Q. From equations A.35 and A,36, 182 2 + r f l + coBh 2v) 0(n ) 8 1 - cosh 2y c But, i , ._ , 2 1 + cos 0 1 + tanh y ______£ cosh 2y = 2 = 2 1 - tanh y sin 6 Thus, sin 0 + r «/ 3n 2____ + Ofr ) 2 c • COS 0 E q ■ f (tan 2 0Q)(1 + r) + 0(n3) Hence, equations A*39, A.40, and A.41 become ® " o “ 'fCtan z 6 o ) (1 + r) [cos 2pt + 4 o (se c 2 0 o ) (1 + r)(cos 4yt - cos 2 0Q)J + 0(n3) (A.42) Y + Mt + y (1 + r ) (2 + tan 0 ) (sin 2y t) O h o 1 + ^(1 + r) (2 + tan 2 0Q) x [2 cos 2yt k2v*i sin O + 0(n3) (A. 43) , 1 + cos 0 183 ♦ ■ ♦o[jr* ^ o s - r ^ ( 1 + f (1 + r)<2 + tan2 e,)) 1 o + 0(“3>J * ■ f CC08 e^(8lno 2"t>(1 + 4(l + r) (2 + tan2 0 )coa 2yt) + 0(n3) (A.44) where I 3 1 + r ------I 3 " 1 e < 0 if I3 < I e >, 0 if I3 > I 0o * I * The only problem now remaining 1 b to find the angular rate, y, in terms of e, 0q, and r. Again from Whittaker [ref. 46] (I - I ) (I c - h2) ( y ) - — - ---- — + 2q + 0(q4))4 Substitution of the expressions for s, q, and r into the above equation' gives h e2 2 (r + i)! - o tan4 6 o (1 + r))] + 0(n3) . (A.45) 184 Hence, the coefficient of t in equation A.44 becomes £ [1 + e2(l - i(l + r)(2 + tan2 0o))] + 0(n3) . Case 3.—Symmetrical Body and a Small Constant Torque For the purposes of this analysis it will be assumed that the only torque existing Is a constant torque, L, directed along the principal spin axis kg. The torques Lp, Lp, and may be expressed as ■Li 3 _ ' V (A.46) LJ3 “ (♦)(0)(4r) LJ6 -Lk 3 “ -Lk6 " where ’ 0 ‘ *L1 6 ' - 0 (A.47) Lj6 -h c fi- . h - Then ” L sin ♦ sin 0 * m -L cos ♦ sin 0 (A.48) Lj3 .L . L cos 0 Now, for small 0, equations A.17, A.18, A.19, and A.20 become 185 I*9 cos 6 h sin 0 0 sz - T - sin h fa L h 8 , , I " "h* cot g 6 h > (A. 50) (*-+) cos 6 If we further assume that L Is also very small, and all terms containing the factor LQ are negligibly small, we have " <5 “ 0 e 4- 0 • h L (A. 51) fa * h/1 $ 0 m / I" I 3 \ _ - _h cos 0 1 )- Note that the term cos 6 is retained in equation A.51. This is done in spite of the small angle approximation for 0 in order that this equation 186 and equation A. 21 obtained for Caae 1 w ill have a sim ilar appearance except for h. From equation A.51 h * Lt + h o r 2 * - ^ - + i ot + *o ) (A. 52) " V 2 • * - - ^ * 1 ------cos 0t + *ot + i,o where “in h - (A. 53a) o cos o h 4 - (A. 53b) O X and A o " ~T~1 it— . cos ® o • (A.53c) From equation A.23c “3 " lJ + W3o * *(A,5*) Also, from equation A.10, 2 » + coa 6$ . (A. 10) I 187 For the approximation of equation A.51 L u>3 - h COS 0 + Y C°S 0 (A.55) or « — cob 0 + u, (A.56) From either equation A,54 or A. 56 It Is clear that may be approximately expressed as a lin e a r function of tim e. The amount of change In for any specified time interval and fixed 1^ Is dependent only on the magnitude of the constant torque L. The material presented in this appendix must be regarded as well known, and has been presented here to provide a complete background for the assumption made for the dynamical model. The material of this appendix also provides necessary background to explain or describe anomalies observed in the flight data. APPENDIX B STAR MAPPER TIME DETECTION CONCEPT This appendix presents a discussion of the system environment and parameters which lim it performance of the s ta r mapper* Symbols appear ing in this appendix are listed and defined in a glossary of symbols attached as the last section of the appendix. Star mapping is accom plished by optically scanning a set of stars distributed about the equator of a spacecraft. The spacecraft considered here is a spin- stabilized vehicle. The spinning motion of the vehicle causes star Images to pass over a reticle which is configured to generate two groups of coded pulses at the output of the sensor. The amplitudes of the signals out of the optical system w ill be proportional to the spectral radiance of the scanned stars, and thus classification of stars accord ing to their visual magnitude is possible. The time of the occurrence of the pulse groups is related to the separation angle (azimuth) between stars, and the time separation within pulse groups is related to the star's elevation angle. Principle of Operation The basic star mapper consists of a telescope, a reticle, and a photomultiplier tube. (See Figure 30.) The telescope is mounted with the optical axis normal toi the spin axis of the vehicle. The reticle is centered in the focal plane of the telescope and the photomultiplier is mounted behind the reticle. The reticle is opaque with two groups 188 189 Spin axis (azimuth is measured about spin axis) Secondary mirror , Primary mirror * Elevation! Star angle PhotomultipLler Reticle Radio transmitter I Note: Lines represent transparent openings in an opaque reticle Reticle-1 Elevation H d h - Spin motion (azimuth) Figure 30. Simplified Drawing of Star Mapper. 190 of transparent slits. The slits of one group are parallel to the spin axis of the vehicle and the second group is placed at a known angle to the axis. As indicated in Figure 30, the spinning motion of the vehicle causes the star image to pass over the reticle, and consequently, the input to the photomultiplier will be a series of coded pulses of radient energy for each group of slits. The number of pulses 1 b dependent upon the number of transparent openings of the reticle. The time of occur rence of the two pulse groups provides a measure of the azimuth angle to the star, and the time separation of a single pair of pulse groups provides a measure of the elevation angle to the star. The amplitude of the pulses of a particular pulse group provides a measure of the intensity of the scanned star. The output of the photomultiplier tube is telemetered to a ground station for data processing to determine the attitude of the vehicle. System Description The star mapper concept illustrated in Figure 31 is briefly described in this section. The star signal which is scanned by the telescope is dependent upon the spectral radiant intensity of the star. Telescope parameters that affect this signal are field of view, which determines whether or not the star is viewed; aperture of * the telescope; spectral characteristics, which limit the spectrum of . acceptable energy; and efficiency of the optics, which determines the efficiency of transfer of energy to the focal plane of the telescope. The combination of the reticle and vehicle spin causes the signal at the detector to appear as a sampled signal. The signal as sensed by the photomultiplier is also a function of the spectral response R eticle Airborne Scar signal Telescope Photo (sampler) m u ltip lie r electronics Transmitter S tar map S tar Receiver Signal detection & Comparator time of detection generation identification Time Reference s ta r map Figure 31. Simplified Block Diagram of System. 192 characteristics and gain of the photomultiplier tube. The airborne electronics operate on the signal with a gain compatible with trans mitter requirements and a band pass compatible with the sampling fre quency produced by the combination of reticle and vehicle spin. In the ground data handling equipment the signal amplitude Is com pared with a variable threshold value for signal detection and classifi cation purposes. This classification will also include time of occur rence of the star signal and time elapsed between repeated scans of the same star. The elevation angle to any scanned star may be determined from the elapsed time between successive pulse groups produced by the star. These times may be converted to angular measurements through a knowledge of the vehicle spin rate and the geometry of the reticle. This tlme-angle relationship is illustrated in Figure 32. The time of occur rence of a pair of pulse groups is indicated by t^, t2» ...» t^, the time lapse between a pair of pulse groups is indicated by At^, At2, ..., At^, and vehicle spin rate is noted as u. The aximuth angle between two successive stars is defined as “ tk^w° and t*ie e^-evat^on angle of a particular star in the field of view is approximated as (w At^ - 2d) tan (90° - 0)°. More exact relationships are given in Chapter 2. Data Processing • The time of occurrence of a pulse group generated by a scanned star is determined through the process shown in Figure 33. The signals from the receiver pass through the variable-threshold circuit which produces a star presence or star absence signal. This binary signal is then compared with a reference code (a replica of the reticle code) in the decoder. When the proper code sequence is recognized, the time Scar 2 S tar 1 S ta r 3 Star 1 n n n n n p _ T_ —1 At2 |«— At3 |a— c2 Azimuth 1 I Azimuth 2 Azimuth 3 ( t 2 - t i ))u M - ( t 3 - t 2)u (t4 - t 2)u T V ertical field of view ♦. I 2 ■ +- Azimuth 1 ►p Azimuth 2 —I Azimuth 3 ' I Figure 32. Incoming Signal and Computed S tar Map. Reference - code Clock V ariable Coded Digital Receiver threshold signal recorder Analog-to- Clamp digital converter Figure 33. Data Processing Block Diagram. 195 of this recognition Is recorded In digital form, and the pulse amplitude Is also recorded by activating the analog-to-digltal converter at the time of pulse-group recognition. Adjustment of the threshold provides a convenient way of restricting the response of the system to the brighter stars In the scanned field. The result of this data processing Is a map of the scanned star field with classification of the magnitude of the stars and a measure of their azimuth and elevation angles in the scanned field of view. Sensitivity and Noise Consideration i_ , The performance of the system Is best described by its ability to d etect an adequate number of s ta rs and by the accuracy with which the orientation of selected spacecraft axes are determined. The orientation accuracy is a function of angular resolution of the system, whereas the detection problem is a function of system noise and signal characteristics and the number of av ailab le s ta r s . Desired Characteristics For positive identification of a scanned star field it is necessary to Insure that an adequate number of detectable stars will be scanned in a given star field. Figure 34 represents the approximate number of stars brighter than a given magnitude, In a field of view of 1 square degree [ref. 48]. For example, if a 6° vertical field of view is selected, the number of stars brighter than a given magnitude in the resultant scanned star field of 2160 square degrees for a complete vehicle rotation is as shown in Figure 35. It can be seen from this figure that in order to encounter at least three stars per scan of the Number of stars per square degree brighter than E 100 000 10 1000 .001 100 1.0 000 0 2 iue3. Spatial Density Stars..of Figure34. 4 6 en aatc aiude d latitu Mean galactic iul antd, m magnitude, Visual 8 10 lci equator alactic G 12 lci pole alactic G 16 1814 20 196 iue 5 Spatal niy f as e Rvlto o a 6° a of Revolution per tars S of ensity D l tia a p S 35. Figure Number of Stars per Revolution Brighter than a 100 500 0 tca Fil o View. of ield F al rtic e V 1 en actc tic c la a g Mean att e d titu la 2 sa Mgiue m Magnitude, isual V lci equator alactic G 3 4 lci pole alactic G 5 6 197 I 198 mean galactic equator, stars with a visual magnitude of +3 or brighter must be detectable. Ideally the system would be designed to detect, or accept, all scanned stars brighter than the selected magnitude and sharply reject stars which are dimmer than this magnitude. This approach would Insure detection of an adequate number of stars for attitude deter mination, reduce the number of false alarms due to system noise, and reduce the number of candidate stars considered In the identification of the stars In the scanned star field. Telescope and Detector Characteristics A star signal as generated by the photomultiplier is dependent upon the spectral radiant intensity of the star source and the spectral char acteristics of the optics and photomultiplier. The spectral radiant intensities of star sources were determined by using source-temperature estimates and assuming blackbody spectra for stars with a visual magni tude of +3 or brighter. A study was performed to determine representa tive source temperatures for the visual magnitudes of interest. The results of this study indicated that the majority of these stars have source temperatures of 6000°K and higher [ref. 48 and 49]. In addition, the "average" source temperature of the myriad of stars which make up the sky background is 6000°K or lower [ref. 48 and 50]. The representa tive source temperatures clearly indicated that the peak spectral response of the optics and photomultiplier should lie toward the short- wavelength region relative to the visual spectrum. Selection of the telescope aperture and photomultiplier tube gain Is not discussed here, but nominal values required for suitable signal amplification are assumed. 199 Signal and Noise Characteristics The principal problem involved in a system of this type is signal detection in the presence of system and external noises. Before pro ceeding to the problem of signal detection it is of interest to examine the characteristics of signal and noise as sensed by the photomultiplier. The noise sources considered here are: external noise due mainly to scattered and direct starlight, and galactic light; noise of the photomultiplier tube; and electronic noise of the system. The noise due to starlight is considered to originate from the dimmer stars, that is, the nonpromlnent stars. Galactic light is emitted starlight or starlight scattered by interstellar dust in the Milky Way. This external light is considered to be of a diffuse nature, and hence (he noise as sensed by the photomultiplier is dependent upon the effective field of view of the telescope. Effective field of view is defined here as the total clear opening of the reticle in the focal plane. Since the release of electrons from the photocathode is related to the quantum efficiency or to a statistical process, it is important to establish the type of distribution associated with this photocathode current. The Poisson distribution agrees with experimental results and is generally assumed. By definition of the system inputs, the optical * parameters, and the photomultiplier tube, signal detectability may be examined in terms of the number of electrons due to signal and noise at the photocathode. The noise considered here w ill be background noise, photomultiplier dark-current noise, and signal-induced noise. In general, noise sources such as the electronic noise of the airborne equipment and radio-frequency noise are negligible as compared with the 200 other noise sources mentioned here. Stellar background pertains to the quantity and quality of the stellar Irradiation. Quantity refers to the number of stars and their radiance. The quality of irradiation refers to the spectral distribu tion of the stellar radiance. Since the total nuober of dim stars is immeasurable, the irradiance must be discussed in terms of the distri buted radiance of the celestial sphere. The background light as viewed by the telescope may be considered as an equivalent signal in terms of total integrated starlight expressed in effective number of tenth- magnitude stars per square degree of effective field of view [ref. 48 and 50]. This equivalent signal may be handled in a manner similar to the star signal to determine the effective noise level as sensed by the photomultiplier. In the determination of the number of photocathode electrons (due to signal and noise) which are produced in a given interval of time, certain system parameters must be defined. The following three equations express the average number of electrons per unit sample time for signal, S, background, Ng, and dark current, N^, respectively: S - K.K AG - f(m) 1 2 w (B.l) _ 2 Nb - KlK2K3 AG en (B.2) (B.3) The total expected signal in a unit sample time is defined as ST - 5 + Nfi + Nt (B.4) 201 The symbols of equations B.l through B.4 are defined In the glossary at the end of this appendix. Coding and Signal Detection This discussion considers the enhancement of detection through the use of multiple transparent slits in the reticle, where each slit will have a width defined by the angular resolution requirement of the system. By arrangement of the slits in a psuedo-random pattern, this technique may be extended to provide a more nearly ideal discrimination threshold. The output of the photomultiplier will not be a pulse-coded modulation of the star signal. The nature of this signal Is a series of pulses as generated by the passage of the star image over the pair of coded slit groups in the focal plane of the optical system. The amplitude of these pulses, as discussed earlier, is a function of the brightness of the star and the response of the optical system. The widths of these pulses are determined by the angular width of the transparent slits of the reticle and the spin rate of the vehicle. The principle used here is to correlate a reference code (identical to the code of the reticle) with the output of the photomultiplier. When the correlation process indicates sufficient agreement between the reference code and the coded signal» there w ill be an Indication of signal detection with a confidence level determined by the acceptable level of correlation. An illustration of the slgnal-detectlon technique is shown in Figure 36. In this figure the sampler represents the pulse-coded modu lation effect of the coded reticle and the spinning vehicle. The func tion of the amplitude selector and T^ (first threshold) Is to select or gate signals which are greater than a preselected value. The function Dark-current noise Output if Star signal S tar Amplitude plus — scanner Sampler background se le c to r lig h t Decoder or c o rre la to r Output i f Reference code 202 Figure 36. Block Diagram of Detection Process. 203 of the decoder or correlator is to compare a reference code (a binary replica of the reticle code) with the coded signal. The symbol r In this figure represents the number of agreements In a given code sequence. The function of the comparator and (second threshold) Is to act as a gate to permit acceptance of a preselected correlation or agreement count as an indication of a signal representing a star of a given magni tude or brighter. If the characteristics of the signal and noise are known at the input to the amplitude selector and the first threshold Is defined, the probability P(S^ £ Tj) of accepting the signal level of any given sample at the amplitude selector may be determined. With this probability determined, the probability of the decoder threshold being exceeded, P (r >, T2), may be computed. In the detection technique under consideration a reference code of L elements is assumed and the code is defined as a two-level binary function. The code is further defined to contain an equal number of elements of each state, such as M ones and M zeros. The time duration of a single element is determined by the sampling Interval. In the preparation of the coded signal for the decoding process the amplitude selector and the first threshold convert the signal to two levels; that is, each signal level in a sample Interval exceeding T^ w ill be (feslgna- ted as a 1, and otherwise the signal level of that particular sample Interval will be a 0. The distribution of the number of electrons in a sample Interval has previously been described as a Polsson distribution but for ease of analysis the distribution hereafter will be treated as a normal 204 distribution, with Its first- and second-order statistics fit to those of the Polsson distribution. Inaccuracies Introduced by this assumption w ill be considered negligible so long as the expected nunfcer of electrons per unit sample Is 100 or greater [ref. 51J. The assumed amplitude distributions for the noise-only and slgnal-plus-nolse samples are Illustrated In Figure 37. With the aid of the normalized standard varlate Z, the probability that a noise-only sample will result in a 0 Is 1 >?i - z ^ 2 P00 - - J e dZ ( » .« where T - (N_ + N ) Z, - - H . (B.6) A + \ Similarly, the probability that a sample containing signal plus noise will result in a value of 1 is 0 0 -ZZ/2 e * dZ (B.7) Z2 where Ti " ST Z2 ------. (B.8) Now Pqq and P11 are defined as the probability of agreement of a single sample pair of noise and signal samples, respectively. It Is now assumed that each sample of the photomultiplier signal Is independent of any other sample. With this assumption, the i Probability density of number of electrons Figure 37. Assumed E lectron D istrib u tio n fo r Noise-Only and Signal-Plus-N oise Samples. Samples. oise Signal-Plus-N and Noise-Only r fo n tio u istrib D lectron E Assumed 37. Figure Ec Cs is Asmd o b Nral Diti ed.) d te u istrib D Normally be to Assumed s i Case (Each T « % tnad viton: n iatio ev d Standard vrg vle N_-hL value: Average ieol sample oise-only N 1 (r abi hat i - usnoie sml wil xed Tj) exceed ill w sample ise o s-n lu l-p a n sig t a th y t i il b ta (Pro P11 0 0 ? pr lt hat os-ny ape l nt xed Tj) exceed not ill w sample noise-only t a th ility b a b ro (p ubr f ecr s n ctro le e of Number B T Sadr deviation:-^]S"Wg+NT ^Standard vrg vle S-*^4NT value: Average S ig n al-p lu s-n o ise sample sample ise o s-n lu al-p n ig S # 1 o to t s 206 distribution of successive correlations of M sample pairs Is described by a binomial distribution. The probability of i agreements In M sample pairs representing noise-only samples is „ t * v>\ - Ml i , . vM-i 00 (M - i ) I I I P00 ^ " P00* (B.9) Similarly, the probability of j agreements out of M sample pairs repre senting signal plus noise is Pn (J*M) “ (m - j)Ij! P11J(1 " P11)M ^ * (B.10) The sum of noise pair agreements and signal pair agreements is defined as r - i+j . (B.ll) The probability that r will be greater than or equal to some pre selected nuober of total agreements may be determined from T „-l M M P(r ■ Tj) ■ 1 - P(r < - 1 - (B.12) i+T-0 i-0 1-0 where Tj is the acceptable level of agreements. Once P (r ■ T2 ) has been determined for a range of values of .star magnitude, for a range of code lengths, and for assumed noise levels, this detection parameter may be combined with the probability of exis tence of star magnitudes in a single scan of the vertical field of view. This combined or joint probability function is defined here as the system probability of response PR(«) to all stars is a single scan. A system figure of merit used in this study is 207 yPR(m)dm m R - — ------(B.13) / PR(m)dm 00 which Is a measure o£ the eyatem response to stars brighter than m^ compared with the system response to all star magnitudes. Ideally this parameter would be equal to 1. This discussion was based upon several assumptions which are lis te d here In a summary: (a) The azimuth and elevation code groups are Independent, but background noise Is a function of the open slits in both code groups. (b) The sample intervals of each code group are independent. (c) The electrons produced at the cathode of the photomultiplier constitute a Polsson distribution. (d) The shape of the spectral response of the photomultiplier is that of the Btandard eye. (e) Signal and noise from a l l sources are independent random v ariab les. (f) Electronic noise such as Johnson noise Is negligible compared with background noise, dark-current noise, and signal-dependent noise. » (g) The p ro b ab ility th a t more than one s ta r of magnitude brighter than m^ (threshold magnitude) w ill occur simultaneously In the telescope field of view Is negligible. Theadvantages of coding lie principally In the enhancement of detection capability through multiple signal pulses for a given star Image. The described technique will provide this advantage and yet 208 retain the precision of angular resolution of a single-pulse Indication of a star image. However, as code length la increased the system noise will Increase proportionally, since the effective background noise as sensed by the photomultiplier is proportional to the effective field of view, which in turn is proportional to the number of transparent coded openings in the r e tic le . The optimum number of code elements for a given system is dependent upon the noises of the system and the desired minimum de tectable star magnitude. A second disadvantage in coding results from using very short code lengths. This disadvantage is the inability to design an unambiguous code group, that is, a code group which will result in a cross-correla tion function with a uniquely defined peak. The question of deslrabll- * ity of coding in a given system must be answered through an analysis which Includes the particular system parameters and environment. Performance Estimates of a Specific System Assumed System Parameters and Noise Conditions A sample design is presented here to demonstrate the effects of increasing code lengths on system performance. Certain assumed noise conditions and fixed system parameters are listed below: 2 (a) Telescope aperture is 125 centimeters (b) Telescope field of view is 6° by 6° (c) Field of view of a basic transparent slit of the reticle is 6° by 0.015° (d) Luminous s e n s itiv ity of photom ultiplier cathode is 65 micro amperes/lumen 209 (e) Spectral response shape o£ photomultiplier Is that of a standard eye (f) Photomultiplier dark current (equivalent tube Input) Is -13 5 x 10 lumens (g) Spin rate of vdilcle la 2706/second (h) Star density Is assumed to be that which would result from a 6° x 360° scan of the mean g a la c tic la titu d e (1) Background light Is the equivalent of 160, 500, and 1000 tenth-magnitude stars per square degree. In the construction of the signal and noise models the expected values of signal and noise photoelectrons were computed as a function of the assumed system parameters by use of equations B.l through B.4. Visual magnitudes ranging from zero to six were assumed for the star signals. Code lengths were varied from two to 24 elements. In the calculations, T 2 wsb set at 0.75L for all code lengths except L « 2. For this value of code length, T2 was set at 1. For each calculation the value of used for each noise case was the value that would result in a 90 percent probability that 50 percent or more of all slgnal-plus-nolse samples In each code group would exceed this threshold value with the signal assumed to be a star of +3 visual magnitude. For L « 2, the percentage was set at 100 rather than 50, since a single sample must be detected. With this consideration, values of P ^ were determined from equation B.10 for various code lengths by using the three assumed background noise conditions and a signal repre senting a third-magnitude star. From these values of P ^ the required values of were determined by use of equation B.7. 210 By using these first threshold values, P ^ nay be determined for any signal-plus-noise combination and likewise may be deter mined for any noise-only condition. After and Pqq have been deter mined for a set of signals and an assumed noise case, P(r “ T2> may be determined from equation B.12. A representative plot of P(r ■ T^) versus sig n a l le v e l (s ta r v isu a l magnitude) is shown in Figure 38 for various code lengths and an assumed background noise of 160 tenth- magnitude stars per square degree. System Probability of Response PR(n) System probability of response °ay determined by a know ledge of ?m(r “ T2^ ^or var^ous atar magnitudes and a knowledge of the probability of occurrence of a star of a given magnitude P(m). For a scan of 6* by 360° the probability of occurrence P(m) of a star of a given magnitude was estimated for the mean-galactic-latltude curve of Figure 35. This curve was numerically differentiated to obtain the frequency of occurrence of stars of a given visual magnitude in the selected 2160 square degree sector of the celestial sphere. The proba b ility of occurrence P(m) was approximated by dividing the frequency of occurrence of a given magnitude by the total nunfcer of stars N in the 2160 square degree sector. The probability of system response to a star of a given magnitude may then be determined from the relationship PR(m) - P(m) Pm(r - T2> . (B.14) Figures 39, 40, and 41 represent NPR(m) plotted against m at various code lengths and for the assumed background noise conditions of 160, 500, and 1000 tenth-magnitude stars per square degree, respectively. Visual magnitude, 3.75 3.25 3.0 4.0 3.5 .00001 iue 8 Pr lt f tcin s Fnto o Viul Magnitude. isual V of Function a as etection D of ility b a b ro P 38. Figure .001 oba lt f ecton, (r 5 T) percent T«), 5 r ( P , n tio c te e d of ility ab b ro P Bcgon o 10 et-antd Str pr qae Degree.) Square per tars S Tenth-Magnitude 160 of (Background .01 .1 * m 10 301 50 70 90 211 I Figure39 Frequency of system response NPn(m) for four values of L and frequency of occurrence NF(m) Frequency ofSyBtem .Response asa Function of Visual 10 8 0 4 7 2 3 5 9 6 1 Magnitude, Backgroundwith a of160 Tenth-Magnitude Starsper Square Degree. iul antd, m magnitude, Visual NP(m) 212 213 NP(m) 01 to OJ O,8 4) 8 to U uV u a § 8 to>% o VM (J>4 O (3 ^ « &s 9 au S'S' t n BO Visual magnitude, m Figure 40. Frequency of System Response as a Function of Visual Magnitude, with a Background of 500 Tenth-Magnitude Stars per Square Degree. i 214 NP(o) S s O u-l u a u* cr Visual magnitude, m Figure 41. Frequency of System Response as a Function of Visual Magnitude, with a Background of 1000 Tenth-Magnitude Stars per Square Degree. 215 A qualitative measure of system performance may be determined by examination of the relationship between system response to stars of magnitude brighter than third magnitude and system response to stars of all magnitudes. This relationship in ratio form Is given by equation B.13. This ratio Is plotted in Figure 42 as a function of code length for the three assumed background-noise conditions. The system shows continuing improvement with increasing code length for the noise conditions and values of L considered, but the rate of Improvement decreases for code lengths in excess of four elements. For the assumed problem, a good choice of code length would be about eight elements, as more elements do not greatly Increase R but do greatly Increase the complexity of design. This value of code length was selected for the star mapper reticle design. The selection was based on the progressive ly diminishing Improvements for larger code lengths and the Increasing complexity In the Implementation of the code and decoder. iue 2 Sse Fgr o Mei Vru Cd Length. Code Versus erit M of Figure System 42. Figure System figure of merit oe egh nme o ee ns, L ents), elem of (number length Code 1000 A e sur degree square per 0h antd st rs ta s magnitude of number 10th in Background GLOSSARY OF SYMBOLS FOR APPENDIX B 2 effective telescope aperture, centimeters azimuthal angular width of a group of coded silts a s t e ll a r constant approximated by 2»5~m photomultiplier sensitivity, amperes/lumen number of agreements in M noise sample pairs number of agreements in M sig n a l sample p a irs a stellar constant for a star of zero magnitude, 2.1 x 10 ^ lumens/centimeter^ a constant, ------electrons/ampere-second 1.6 x 10"A* equivalent nunber of tenth-magnitude stars per square degree of the effective field of view of the telescope code length nuober of l's or 0's In a code sequence stellar visual magnitude visual magnitude of the minimum detectable star total number of stars in a selected 6° by 360° scan of the celestial sphere average number of electro n s produced per u n it sample time by background average number of electrons produced per unit sample time by phototube dark current equivalent number of reticle transparent openings of width a probability of occurrence of a star of visual magnitude m probability of detection in the caae where a signal is present and probability of false alarm in the case where noise only is present for any star 218 > P(ST • T.) probability of accepting the signal level of any given sample at the amplitude selector Pm(r • T2) probability of detection in the case where a signal Is present and probability of false alarm In the case where noise only Is present for a star of magnitude m Pj^(m) system probability of response Pqq probability that a noise-only sample will result In a aero probability that a sample containing signal plus noise will result in a value of 1 Pqq (1,M) probability of 1 agreements probability of j agreements R system figure of merit r nunber of agreements In a correlation of a given code sequence S average nunber of electrons produced per unit sample time by a source star total average number of electrons produced per unit sample time amplitude detection threshold T2 correlator detection threshold t^ time of occurrence of pulse groups (1 ■ 1, k) At^ time lapse between a pair of pulse groups (1 » 1, ..., k) Z a random variable with zero mean and standard deviation of 1 normalized value of In presence of noise only Z2 normalized value of In presence of signal and noise a width of a reticle transparent slit, degrees m/u unit sample time or observation time, seconds 8 the angle between the two groups of coded silts, degrees vertical field of view, degrees equivalent dark current Input, lumens vehicle spin rate, degrees/second • APPENDIX C STAR MAPPER DESCRIPTION AND SUHIAKY OF DESIGN CONSIDERATIONS The contents of this appendix are Included In this dissertation to provide the reader with a brief description of the design considerations of the star mapper used on the two data gathering flights discussed In the text. It aust be emphasised here that the material presented In this appendix la the result of the efforts of a ten-nan project team of the NASA Langley Research Center. Particular thanks Is given here to the sta ff members of the Flight Instrumentation Division who provided support In the design, analysis, testing and calibration of the Instru ment described In this appendix. The star mapper concept of scanning a star field Is Illustrated In Figure 43 where the star mapper Is shown mounted with Its optical axis nominally normal to the spin axis of the spacecraft. The star mapper consists of a telescope, a coded reticle, and a photomultiplier tube. The reticle Is centered in the focal plane of the telescope with the photomultiplier mounted behind the reticle. The reticle Is opaque except for two coded sequences of transparent s lits grouped In the general shape of a "V". The spinning motion of the vehicle causes a star image to transit the reticle elite; consequently, a series of pulses of radiant energy w ill be sensed by the photomultiplier for each star transit of the two sequences of transparent s lit s . Each sequence of transparent silts In the star mapper was designed to generate an 8-bit code group In which the presence of a star was Indicated by a one and the absence was Indicated by a xero. The elapsed time (Atj) between a pair of pulse groups produced by a single star Is related to the SPIN AXIS STAR MOTION RESOLUTION: 0.015° RErlCl£s TRANSPARENT SLITS STAR FOV 6° BY 6° MAPPER AR A STAR B A t, ELEVATION ANGLeTUU LfUIR J1L 1 m »A 1 O A ^ AZIMUTH ANGLE H • A t SIGNAL OUTPUT VEHICLE CONFIGURATION 221 Figure 43. Star Mapper Concept. 222 elevation angle of that.star In the field-of-view of the star mapper. The elapsed time. (At^) between pulse groups produced by a pair of stars is related to their aslmuth angle separation. A simplified flow diagram of the star mapper signals is shown in Figure 44. The output signals of the star mapper were used to frequency- modulate a telemetry transmitter. These telemetry signals were trans mitted to a ground station for tape recording. Processing of these recorded signals consisted of four significant operations: time de tection of. apparent star transit signals, manual sorting of the apparent transit times to reduce false alarms, identification of the detected stars, and determination of the spacecraft attitude time history. The recorded star-mapper signals along with range time were pro cessed by a pulse-code detector that identified the transit times for each detected star signal. A variable threshold control was adjusted to limit the nuiAer of detected star signals, and the peak amplitude of the first pulse in the sequence of pulses generated by a star was re corded for purposes of classifying the stars according to their signal le v e ls . Deaign Considerations In order to determine the attitude of a spin-stabilized vehicle through use of celestial observations, at least three star sightings were required for each revolution of the vehicle (see Chapter 6). Based on star distributions in the celestial sphere, it has been determined that a minimum detectable star of +3 visual magnitude (m^) combined with a 6° by 6° field of view scanned through 360° of the celestial sphere will satisfy the three-star-sighting requirement when the density of the S ta r Photomultiplier Airborne Transmitter Signal Tube Bleotronios Nominal Launoh, Conditions Data Reoeiver T»pe Star Identification Recorder Decoder Sorting Program Time A ttitu d e Determination \ 223 Figure 44. Simplified Signal Flow Diagram. 1! 224 stars contained within the scanned field of view approximates the star d ensity of the mean g a la c tic la titu d e [ref* 52]. Launch windows fo r space flights must be selected to insure that the density of stars viewed by the star mapper w ill approximate the star density of the mean galactic latitude. Since the basic star mapper measurement is the time a star Image transits a reticle silt, the Instrument accuracy Is determined by the resolution of time detection of a star pulse. Therefore, In design, it is necessary to consider not only the optical field distortion and reticle inaccuracies but also the optical image quality and electronic system fidelity that affect the star-signal pulse shape. The optical and electronic design requirements are further constrained by the specific flight requirements. For the flights described In this dis sertation these requirements Included an approximate 270 deg/sec spin rate, a day or night operation, a mechanical envelope of 40.64 by 25.40 by 54.61 centimeters, and a mass limitation of 18 kilograms. It was required that the pointing direction of selected axes be determined continuously with an accuracy (lo) of 0.006* to .01*. As a means of insuring that the experiment accuracy requirement was met, a lo accuracy of 0.006* was selected, for sightings of Individual stars. This accuracy of 0.006* Included the effects of optical resolution, reticle slit widths, and the resolution of range-tlme measurements. Ideally, the optical resolution and basic slit width would have been equal to achieve a matched-fliter relationship [ref. 53] for pur poses of detecting the presence of a pulse. Aablgulty conditions In star identification through angular separation comparisons pointed out 225 a possible need for star-signal amplitude measurements for magnitude classification. A slit width of 0.015° and an optical resolution of 0.01° were selected as a compromise between matched-fliter character istics and amplltude-measurement requirements. The resolution of range- tlme measurements was expected to be 10 microseconds or 0.003° when the expected spacecraft nominal spin rate of 270 deg/sec was considered. The expected angular error In sighting an Individual star can be esti mated by computing the root sum square of the error due to range-tlme resolution and the error due to the convolution of the optical resolu tion with a basic slit width of the reticle. The result of this com putation was a lo accuracy of approximately 0.006°. The atar signals generated by the photomultiplier are dependent upon the spectral radiant intensity of the star source and the spectral characteristics of the optics and photomultiplier* The spectral radiant intensities of star sources were determined by using source-temperature estimates and assuming blackbody spectra for +3 visual magnitude and brighter stars. A study was performed to determine representative source temperatures for the visual magnitude of interest. The results of this study indicated that the majority of stars of +3 visual magnitude and brighter have source temperatures of 6000°K and hotter [refs. 48 and 49]. In addition, the "average" source temperature of the myriad of stars which make up the sky background Is 6000°K or colder [refs. 48 and 50]. The representative source temperatures clearly indicated that the peak spectral response of the star mapper had to lie toward the short wave length region relative to the visual spectrum. Under, the consideration of available glass for the optical system and photomultipliers, the short wavelength cutoff of the Instrument was selected to be 3200 226 angstroms as a design goal. The long wavelength cutoff of the Instru ment was selected to be 6000 angstroms to provide a sufficient amount of star irradlance integration for star-signal detection and to reduce the response of the instrument to background light. The design of a star mapper unit followed a series of tradeoffs between optical transmission, detector sensitivity, optical distortion, reticle design or curvature, background noise, electronic bandwidth, detector dark-current noise, vehicle spin rate, resolution requirements, size and weight (mass). The star mapper design is summarized in the following section* A more detailed description may be found in [ref. 8]. Design Elements A ruggedlzed photomultiplier tube having an approximate S-ll spectral response was selected as the photodetector. This tube had -15 a dark current of 1.5 x 10 ampere, had a responslvity of 65 micro- g amperes per lumen, and used 14 dynodes to develop a gain of 10 at a nominal supply voltage of 2800 volts dc. The gain stability character istics of the tube Introduced a stringent stray-light requirement, since when placed in a system capable of detecting a +3 visual magnitude star, a -3 visual magnitude star would cause the tube to exhibit gain degra dation; and the tube was virtually destroyed by exposure to light levels brighter than -5 visual magnitude. In design of the optics, both reflective and refractive systems were investigated. With a reflective system, it was not possible to take advantage of the space available to provide an effective sun shield. Catadloptric systems were rejected because of the large central obscura tion resulting from the secondary mirror. The refractive optical system 227 chosen for the star mapperwas a modified Petzval design with a 38.1- centlmeter focal length and a 12.7-cehtimeter clear aperture [ref. 8]. The design provided a maximum blur circle of 0.01° over the field of view. The measured spectral response of the entire system (optics and photomultiplier) for the star mapper Is shown in Figure 45. The basic slit width of the reticle is a function of the desired resolution and In the star mapper was made to be 0.015*. The reticle design of the star mapper can be seen in Figure 46. By observing the group of vertical silts In the figure (from right to left-hand side), it can be seen that the first vertical slit was 0.03* wide with a blank width of 0.015* being next, the second vertical slit was 0.015* with a blank width of 0.045° being next, and the third vertical slit was 0.015* wide. This grouping provided an 8-blt recognition code of 1-1-0-1-0-0- 0-1. The group of slanted slits had an identical recognition code with the narrowest separation between the vertical and slanted code groups being equal to one code group. The choice of the 8-blt code pattern was made to provide a high probability of discriminating stars of +3 visual magnitude and brighter against the expected background [ref. 52]. The output of the photomultiplier tube was amplified by a bandpass amplifier to obtain the desired gain and pulse fidelity. The low- frequency cutoff was required because of slowly varying levels o t s p a tia l background noise. The high-frequency cutoff was a function of time of star passage over the narrowest reticle slit, optical resolution (blur circle), and telemetry system filtering. After allowing for spln-rate variation and frequency degradation by signal processing equipment, the high-frequency response was selected to be a three-pole filter with Relative response 100 30 10 50 90 70 3600 4000 iue4. Measured Spectral Responsethe of Star Mapper Figure45. 4400 805200 4800 aeegh angstroms Wavelength, 5600 6000 6800 228 4.011 2 6 2.0048 1 6 16 - 0.010 centimeter (.015*) 2.005 1 & 4 L 3 fi NOTES: 1. Transmissivity of clear area in slits - 95% 4.011 Transmissivity of opaque area less than 0.1% 3. Dimensions in centimeters 4. Slit widths are shown exaggerated for clarity. ro \oK> Figure 46. Reticle Configuration. 230 a -3 decibel point at 33 kilohertz. The low-frequency response was designed to be -3 decibels at 15 hertz with a slope of 6 decibels per octave. The telemetry system required that the maximum output of the star mapper be 5 volts. To Insure that this maximum output of 5 volts was not exceeded, a dynamic range of 0.256 to 4.0 volts was set for +3 to 0 visual magnitude stars of the AO spectral class (11 000°K), respec tiv e ly . Each power supply was s e t fo r a photom ultiplier gain of 10*\ The electrical components (high-voltage power supply, photo m ultiplier, and bandpass amplifier) were interdependent and required calibration as a unit. In order to assure recognition of a +3 visual magnitude AO star, It was required that the signal levels produced by stray light be less than or equal to the signal level produced by a +4 visual magnitude AO star. For sunlight, this specification required an attenuation factor -12 of 10 for all rays entering at angles of 25° or greater from the optical axis [ref. 8]. Because of this requirement and because of vehicle space constraints it was necessary to use folding mirrors. With the addition of these folding mirrors, the overall optical effi ciency, Including all elements, was approximately 60 percent. Calibration and Testing A field of view mapping test was designed to determine the effects of optical Inaccuracies on angular position measurements In the field of view. The basic method employed to determine these inaccuracies was a reconstruction of the reticle through star mapper measurements and comparison of this reconstructed reticle with the reticle design. 231 The star mapper vaa rotated so that Its field of view scanned a colli mated source. The output and the angular positions of the star mapper were simultaneously recorded. During the scanning process, the Image traversed the openings In the reticle. The output signal represented the convolution of the blur circle with the reticle slits. The reticle was reconstructed by plotting the amplitude of the convolved signal against measured azimuth and elevation angles of the star mapper. The results of these measurements are discussed in Appendix F. The ideal means of calibration for a photometric device of this type would be exposure to several stars of different visual magnitudes and spectral classes. However, knowledge of atmospheric attenuation at any particular time and the presence of scintillation make this method questionable for the short term observations of this system. Therefore, a star simulator [ref. 8] was constructed to calibrate the star mapper. Using this star simulator, the star mapper was exposed to three star spectral classes (AO, GO, and MO). For the AO star with visual magnitude of 0, the system gain was adjusted until the star mapper output signal reached a value of 4 volts. For each spectral class, the irradlance was varied from an equivalent star with mv ■ 0 to a star with my - +3. The slgnal-to-noise ratio (exclusive of back ground noise) was found to be approximately 13:1 for the stars with a visual magnitude of 0 to the AO class. A summary of the calibration of the star mapper is shown in Figure 47. Transit Time Detection The data processing required to detect and identify the stars sighted by the instrument is briefly described. Additional discussion i- i Output Signal,V olta 0.2 0.4 2.0 .0 4 0.5 iue4. Star CalibrationHapper Figure47. 1.0 sa Magnitude isual V 1.5 2.5 tr Type Star M - AO GO 3.0 233 of this data processing may be found In [refs. 8 and 9]* As previously discussed, a single-pulse sequence for a particular star transit con sisted of an 8-bit code pattern. The code pattern produced by the reticle slit design was 1-1-0-1-0-0-0-1. Since the reticle Is opaque except for its transparent slits, the next bit In time would be a zero. This feature was UBed to add another zero bit to the code group since a return to zero situation was desirable In the decoding operation. In order to Improve on the time resolution of a detected star transit, each bit of the code sequence was subdivided Into four elements. With these changes, a single star transit generated a code sequence of 36 b i ts . A simplified diagram of the decoder is shown in Figure 48. As indicated in this figure, the raw star mapper signals, range time, and two reference frequencies of 1 kilohertz and 100 kilohertz were tape recorded. The 100-kllohertz signal was used as a 10-microsecond vernier during l-milllsecond intervals. The raw star mapper signals were con verted to a two-level digital pulse train by use of an adjustable threshold. This 36-blt, two-level, coded-pulse sequence was compared in a correlator with a reference pulse sequence consisting of an equal nunber of bits* Detection threshold was set for two conditions of logic. * Detection of a star transit across a sequence of transparent slits was acknowledged provided that comparison of the reference code sequence with a given sequence of 9 basic bits produced agreement of any two ones and all five zeros or three ones and four zeros. Acknowledgment of transit detection was in the form of a gating signal which enabled Recorded sign als Gate D ig ita l I Variable- Coded Signal Shift register clock pulse storage T O T — conditionar 36 b it generator signal jmnnE of Correlator Match lo g ic identification Adjustable tra n sit threshold MM Reference code time shift register Gate and 1-kHz Signal s ta r reference conditional Binary coded T declnal counter Gates lOO-kHx Signal signal reference conditional anplitudi: Range tin e hj Figure 48. Decoding of Star Mapper Signals. 235 digital recording of range tine of detection, threshold setting, and peak amplitude of the first pulse in a slgnal-pulse sequence. The peak-amplitude detector vas reset to zero after each code sequence recognition* The variable threshold and the tape recording feature permitted selection of a limited number of stars per vehicle revolu tion. This capability reduced the number of possible ambiguities In the subsequent star identification program. Detection times were set for the trailing edge of each 36-blt code sequence. Detection time resolution was limited by the time duration of one element of the 36- bit code sequence and range-time uncertainties. The time duration of each bit of the 36-bit code sequence was controlled by a variable-clock pulse generator which was adjusted for the best estimate of vehicle spin rate and the angular resolution of the smallest transparent slit of the reticle. The spln-rate estimate was readjusted as attitude determ ination re s u lts became av ailab le [re f. 9]. Results of the transit time detection were stored In a digital listing of star transit times and amplitude of each star signal. Pairing of transit time is aided with use of recorded signal amplitude measurements. The relationship of these transit times and transit time p a irs to the geometry of the s ta r mapper and the r e tic le s l i t planes is discussed in Appendix F. A discussion of the processing of the transit times obtained from flight data is presented in Chapter 7. APPENDIX D STAR IDENTIFICATION PROCEDURE Star Identification nay be done either through comparison of the computed angular separation of pairs of observed stars with the tabulated angular separation of pairs of catalog stars or by comparison of observed slit crossing times of data stars with predicted times for all stars expected to fall within the star mapper field-of-vlew. In the former case, more computations must be done with observed data, but the technique Is somewhat independent of uncertainties In Initial spacecraft attitude. In the latter case, star identification is strongly dependent upon knowledge of the Initial spacecraft attitude, especially the direction of the spacecraft principal spin axis. Because of the large Initial uncertainties In the spacecraft attitude anglea ♦ and q , as discussed in Chapter 4, and no particular desire to achieve a real time solution in this study, the first method of star identification was chosen. However even with this method an estimate of the total vehicle spin rate was needed to make calculations of the angular separations of the observed stars. This estimate of vehicle spin rate was obtained by autocorrelation of the star transit data. A discussion of these angular separation computations of observed stars and their use In star identification Is presented in this appendix. A digital computer program was developed to Identify each star associated with each pair of detected transit times. Special acknowledgement is given here to Hr. D. E. Hinton of the Flight 236 237 Instrumentation Division for his efforts in the development of the star identification program described in this appendix. One of the primary requirements of the program was that the transit times be in chronological order and that the times appear in pairs, each pair being from the same star* The equatorial coordinates of the reference stars used in this program were obtained from the Boss catalog [ref. 30], and updated to the nearest 0.1 day of the time of flight by use of standard astronomical techniques [ref. 54 and 55]. A candidate reference list of stars was selected from all stars falling in the range of +3.5 visual magnitude or brighter. This selection was established on the basis that there will always be at least three acceptable stars in the scanned region of the sky which are brighter than about +3.5 visual magnitude. Of course, the upper limit will also Include the planets. However, this factor should cause no identification problem since a planet in the field of view will cause the star mapper to saturate, and an acceptable pulse tr a in w ill not be generated. The s ta r lis tin g was fu rth e r reduced by selecting all stars falling within a selected band of the celestial sphere. The limits of this band, shown as an extended field of view in Figure 49, were defined to have 2enlth angleB of 76* and 104° away from the direction of spacecraft angular momentum vector as estimated from launch data. This band of +14° about the spacecraft equator was necessary to accommodate the +3° field of view of the star mapper, an uncertainty of +5° in the estimated angular momentum direction, and a possible 6° vehicle cone angle. All stars of the reduced reference list were considered aB candidates for the transiting stars. An additional listing of all these candidate stara was made in the form of a catalog of angular separation g£ of each pair of stars in the reference list. 238 Vehicle spin axis Optical axis Figure 49. Simplified Scan of the Star Mapper Extended Field of View About the Celestial Sphere for the Case of Zero Cone Angle. 239 Star identification waa principally a process of finding a cataloged angular separation which agreed to within a fixed tolerance Ag of an estimated angular separation gQ of a pair of observed stars. The angular separations of the cataloged stars are easily determined from their equatorial coordinates. In Figure 50, the position of star G^ on the celestial sphere is defined by coordinates (a^, 6j) and the position of star Gg is defined by the coordinates (ag, 6g), The Angular separation between and Gg, defined by arc G^Gg and designated as ge, is cos gc ■ sin 6^ sin Ag + cos 6^ cos 6g cos(a^-ag) . (D.l) The angular separation between a p a ir of observed s ta rs was estim ated by use of the times at which the two stars transited the two slits of the reticle. This estimation was baaed on the assumption that vehicle cone angle and the misallnement angles were zero. These assumptions led directly to the definition of the azlmuth-angle separation of a pair of stars as observed by the star mapper; that is, • m n (t — r \ ( D »2 ) e s t v(k+l) vk where w is the a priori estimated total spin rate and the subscripts k G 8 C and v indicate the kth star transit of the vertical slit plane. The azimuthal separation of two observed stars as defined in equation D.2 is analogous to the difference in the right ascension angles of two cataloged stars as shown in Figure 50. The elevation angle e^ of a single star as observed by the star mapper may be written in a manner similar to that of equation F.4 of Appendix F as 240 ■ortta pole Figure 50. Angular Separation of Two Stars on the Celestial Sphere. 241 CD.3) where the subscripts sk and Vk refer to the kth star transit of the slanted slit plane and the vertical slit plane, respectively. The estimated total angular separation gQ of a pair of observed stars say be written in the identical form of equation D.l as cos gQ - sin e^ sin e^^ + cos e^ cos e^_^ cos a . (D.4) A pair of stars in the reference list was selected as the prims candidate for the identification of a pair of observed stars if IBc " ®0I i (D.5) where Ag is some fixed allowable difference In th e two angular separations. This difference was adjustable and was set to be that value which would r e s u lt in the se lec tio n of a su ffic ie n t number of prime candidate star pairs, so that at least three stars per vehicle spin period were identified. After star-ldentiflcatlon processing was completed for one spin period, all information related to the identified stars was a v a ilab le. This information Included the Boss catalog number, visual magnitude, right ascension, and declination of each identified star. The information also Included a listing of transit times of each Identified star for both the vertical and slanted slit planes. Each succeeding spin period was treated in a similar manner with no knowledge from previous spin periods 240 forth pole Figure 50. Angular Separation of Two Stars on the Celestial Sphere. 241 (D.3) where the subscripts sk and Vk refer to the kth star transit of the slanted silt plane and the vertical slit plane, respectively. The estimated total angular separation gQ of a pair of observed stars may be written in the identical form of equation D.l as cos gQ ■ sin e^ sin + coa ^ cos ^+1 C0B a * A pair of stars in the reference list was selected as the prime candidate for the Identification of a pair of observed stars if lgc " 80l < ^ (D.5) where Ag Is some fixed allowable difference In the two angular separations. This difference was adjustable and was set to be that value which would result in the selection of a sufficient number of prime candidate star pairs, so that at least three stars per vehicle spin period were Identified. After star-ldentiflcation processing was completed for one spin period, all information related to the Identified stars was available. This information included the Boss catalog number, visual magnitude, right ascension, and declination of each identified star. The information also included a listing of transit times of each identified star for both the vertical and slanted slit planes. Each succeeding spin period was treated in a similar manner with no knowledge from previous spin periods 242 assumed, since the flight data were expected to contain a minimum of three stars per spin period. After processing was completed for all spin periods, a synopsis was made of all stars identified and the pertinent information for each star was printed. These data were next used in the parameter determination program. It may be recalled from Figure 22 that the star mapper data at the input to the star-ldentlflcatlon program were in the form of star transit time pairs where each transit time pair resulted from observation of a single star. This time listing was grouped in triplets of transit time pairs for each vehicle spin period which was obtained from <*>_„,.•68C This grouping is illustrated in the following example sketch: |- Triplet 1 ------►! Star 1 Star 2 Star 3 Star 4 Star 5 tvl tsl tv2 ts2 fcv3 ts3 fcv4 ts4 Cv5 ts5 |-«------Triplet 2 * | •*------Spin period ------ The f i r s t t r i p l e t i s composed of tr a n s it time p a irs 1, 2, and 3 and the second triplet is composed of star transit time pairs 2, 3, and 4. This grouping into triplets was continued until all transit time pairs in a spin period were exhausted. The angular separation of each p a ir of s ta rs in a t r ip l e t was determined by equation D.4. These angular separations were used to obtain prime candidates for the observed star pairs from the reference list of stars. As many sets of prime candidates were selected for stars 1 and 2, then 1 and 3, and finally 2 and 3 as could satisfy equation D.5. 243 If this is done properly, at' least one of the prime candidates for star 1 of (1,2) and star 1 of (1,3) should be the same. This procedure was repeated to identify star 3 of (1,3) and (2,3), and star 2 of (1,2) and (2 ,3 ). This procedure is illustrated by the following example: Given a computed angular separation of two observed stars, the listing of angular separations of all pairs of stars in the reference list was searched for angular separations meeting the condition of equation D.5. Each pair of observed stars was associated with a subllstlng of prime candidates taken from the reference star list. After this procedure was completed for stars (1,2), (2,3), and (3,1), the following sublists of prime candidates were available: Prime candidate star pairs from reference list for observed pairs — 1,2 2,3 3,1 1.5 5,6 9.1 19,29 9,4 7,17 14,17 129*71 18,22 22,29 30,18 41,42 33,36 The star numbers denoting candidate stars merely refer to the index num bers chosen for the reference list of stars and should not be confused with Boss catalog numbers. Simultaneous comparison of the three lists of prime candidates should yield one triplet of reference stars for the 244 triplet of observed stars 1,* 2, and 3. In the example, It is apparent that the only triplet of reference stars which has common stars in each of the three lists of candidates star pairs is 7, 9, and 29. This comparison results in a tentative identification of observed stars 1, 2, and 3 as reference stars 7, 9, and 29, respectively. Sometimes a triplet of observed stars was tentatively identifiable by more than one triplet of reference stars. The ambiguity was overcome by using the identified stars of each possible triplet and their associated six transit times in a least-squares program to solve for the reduced set of parameters $ + 0 and $. All triplets were rejected when values of 0 and ♦ disagreed by more than +10° from the launch data estimates of these two angles. The triplet which produced the smallest functional residual in the least-squares program was selected as the identified triplet out of all the remaining triplet ^mdldates. Once a triplet was tentatively identified, the Index numbers of the Identified reference stars were stored in a sample table aB follows: Sequence of T rip le t number s ta rs in a t r i p l e t 60 245 Identification of the second triplet of stars Ideally reproduces the index numbers 1 and 16, since the next triplet is obtained by dropping the first star and adding a new one at the end. In the example, the added star is assumed to be index number 38 which yields Sequence of T rip le t number s ta rs in a t r ip l e t If the program was unable to identify the stars in a triplet, a negative number (-1, -2, or -3) was stored in the table in place of a reference star index number, as Indicated in the following list representing the identification of all stars in a complete spin period: Sequence of T rip le t number s ta r s in a t r i p l e t 246 The final atep in the identification process was to examine the reference star Index numbers In all but the last two columns of the table of identified stars for identical index numbers. If two or more index numbers in each column were identical, then positive Identification was assumed for the reference star associated with that index number. If no two index numbers in a column were a lik e , id e n tific a tio n was n o t made. Identification of the first and second star In a spin period required that the first two transit time pairs be updated by a spin period, the star pair representing these modified transit times be repeated at the end of the list, the second and third entries of the next to last column be copied into the first column, and the third entry of the last column be copied in the second column. APPENDIX E RELATIONSHIP OF THE SELECTED ESTIMATOR TO THE SEQUENTIAL LEAST SQUARES AND WEIGHTED LEAST SQUARES ESTIMATOR The estimation theory discussed In this appendix must be considered to be well known and Is available from many sources (e.g. [refs. 56, 57, 58, and 59]. However, the investigation of the application of a sequen tial ■estimation procedure to the problem of this dissertation is considered to be of specific interest. Hence the presentation of the contents of this appendix is considered to be appropriate for both completeness and clarity. The solution for the correction vector AX has been found for the weighted least squares case and is given in equation 3.38 of Chapter 3 as AX(J) - - [(Pf)J x W fJj] 1 (Vo J k V 1*^ • (E.l) For ease of notation, the superscript j, denoting the jth estimate, or the jth iteration, will be dropped in this discussion. In addition, let K(7f)fc - Vfc and 247 248 Then equation E.l becomes l- l “ - - Kr' \ ] i • (E-2) An additional observation will add a row to the V^f R(k)» and the f£ matrices. The value of the correction vector based on the original k observations and the (k + 1) observation becomes AX(kfl) - - : ± ] R*1^ ) [-71]} [vk ! 2LT] R_1(k+1) [tt] where v Is a 1 by m row vector and f' Is the (k+1) observation. Recall ing that R la a diagonal matrix of the variances of the observations, equation E.3 may be written aa AX(kfl) - - [v^R_1(k)Vk + vTav] 1 + / af’] (E-4) where 1 a (E.5) o*(k+l) As noted in Chapter 3, we may define as i - l pk ■ K R'1(k)vJ and i - l ?k+ i - [vkR_1(k)Vk + vTav] (E.6) 249 where end E^i ere m by a me trices. The matrix may be written es Pk+1 - {[V£R'1 pkfi - K +KR'1 Where Im is the n by m identity matrix. Using the definition for in equation E.6 we have Pwa ■ [ Im + Pl / “*] »k • tt.9) It is interesting to note that the matrix inversion of equation E.9 can be avoided by use of a simple procedure. Let A ■ PiV^ak— (E.10) B ■ v where A 1 b a m by 1 matrix and B is a 1 by m matrix. Now -AB £ -AB • i I If (Ij^ + BA)**1 exists, then -AB - -ACI^+BA) (Ij^+BA)-1B (E.ll) where 1 ^ is the Identity matrix of the dimension of the matrix product BA. Now add (I Q + BA) to both sides of equation E .ll Xm “ tJm+AB] -AlI^+BAltl^+BAl^B • (E.12) It should be noted that the dimension of the identity matrix I is m the dimension of matrix product AB. Equation E.12 can be written as l n - [I+AB] - [A+ABA] [ I j^+BA] -1B or I - [I+AB] - [I+AB] A[ItlA+BA]"1B . (E.13) m m d ua The term [I m+AB] may be factored in equation E.13 to give I - [I+AB] [I - A[I_ .+BA]”1 B) m m m da or - lm - AU^+BA]-1 B . (E.14) The dimensions of the A and B matrices cause the quantity [I^+BA] to be scalar since the dimension of the vector product AB is 1 by 1. Equation E.14 becomes [ I ^ r 1 - (E.15) 251 Applying equation E.15 to equation E.9, becomes T P.lei v av — I - (E.16) plc f l * “ U+wPjA) and the correction vector AX Is AX(k+l) - -Pk+1 [v^R‘"1(k) f ’ (k) + vT a f ] (E.17) Substitution of equation E.17 into E.16 gives T P. v av lc^ — AX(k-fl) - - I - PkVkR“1(k) Let T P,kr- v a K - (E.19) (1-K.PiV a) Then AX(k+l) - AX(k) - [KvPj^a - P^ajf* - KvAX(k) . (E.20) The second term of the right side of equation E.20 may be written as [K v P ^ a - P j/a J f * -K f' (E.21) from use of equation E.19. Equation E.20 then becomes 252 AX(fcKL) - AX(k) - K[f ’ + vAX(k)] • (E.22) If we make the assumption that the true vector X is the linear combina tion of the last estimate and the corresponding correction vector we have X - X(k)+AX X - l(W-l) + AX(k+l) • (E.23b) Then X (kfl) - X(k) - [AX(krfl) - 6X(k) ] . (E.24) Substitution of equation E.22 into E.24 results in X(k+1) - ]jt(k) + K [f' +iiAXOi>] • (E.25) It should be noted here that the matrix of equation E.16 and the X(k) vector of equation E.25 are usually updated to the instant of time of the (fcfl) observation before these equations are used. The updating can be denoted by p£ - *(kH,k)Pk *T(fcfl,k) $(k+l,k) X(k) (E.26b) 253 where ♦ represents the transition matrix for the dynamical system. However, the estimation problem considered in this paper is an Initial value one. That is, the values of the X vector are all referred to a selected time instant and are not affected by the total observation time. Thus, the estimate X(k) refers to an estimate of the parameters of X referred to some initial starting time, and is not affected by the (k+1) observation except for a statistical improvement in the estimate. Therefore, the 4 matrices of equation E.26 will be assumed to be Identity matrices in this discussion. It will now be shown that the results developed for the recursive weighted least squares process agree with the sequential Kalman filte r for the assumed dynamical model. The sequential Kalman f i l t e r estim ation algorithm s [re f. 59] w ill be given here w ithout proof. The same assumptions w ill be made fo r the transition matrix in this formulation as for the previous discussion. The equations for the sequential Kalman estim ator are X(k+1) - X(k) + K[f' + VAX (k) ] * T p V a K - ---- — (l-fv P ^ a ) (E.27) Pkfl " lIm - The results given in equations E.27 clearly agree with the weighted least squares recursive estimator for = 0. How is the covariance matrix which results from permitting the dynamic model to be corrupted with a 254 white noise process having zero mean. Since the assumed model is assumed to be well known and noiseless, the covariance may be justifiably set to be zero. For this case, the recursive Kalman estimator and weighted least squares recursive estimator for the assumed model and problem are identical. If it is further assumed that each observation error has equal variance, the weighted least squares recursive estimator reduces to the least squares recursive estimator. For this special case, the three discussed recursive estimators are identical. Since this special case is the one considered throughout the paper, the standard least squares estimator was used. In addition, since data processing time was of no concern, batch processing was used for convenience through out the study. A discussion of the problems associated with the use of the sequential estimator Is given in the chapter describing flight re s u lts and in the chapter describing recommended fu rth e r stu d ie s . APPENDIX F RETICLE MODELING AND MEASUREMENT OF OPTICAL PARAMETERS Since the parameter estimation procedure depends upon the accuracy of the measurement of time Indicative of a star transit of selected slit planes of the reticle, It Is Important to know the effects of optical Inaccuracies or uncertainties In angular position measurements in the field of view. These angular uncertainties may be directly related to uncertainties In time measurements of angular events. Reticle Reconstruction The method used to determine these Inaccuracies was to develop a mathematical model of the reticle through laboratory measurements fol lowed by a comparison of this modeled reticle with the theoretical r e tic le design [re f. 8 ]. Acknowledgement Is given here to the support of Mr. A. Holland and Mr. D. E. Hinton of the Flight Instrumentation Division for their support In the design and implementation of the laboratory test setup and their measurements of the parameters used In the analysis of this appendix. The test setup used for the reticle reconstructio n measurements Is shown in Figure 51. The s ta r mapper was rotated so that Its field of view scanned a collimated source. This * scanning caused the Image to traverse the reticle silts and thus the star mapper output represents the convolution of the blur circle with the reticle slits. The reticle was reconstructed by plotting the amplitude of the output signal against measured azimuth and elevation angles of the star mapper. 255 star mapper Rotary table Folding mirror, Optical encoder Parabolic I and pickup collimating I mirror I \ Output of encoder Chopper and star mapper Aperture to recorder Heutral~denslty filters Spectral filter Imaging optics Lamp Star simulator 256 Figure 51. Reticle Reconstruction Test Setup, 257 The collimator used for this test has a 254 cm focal length and a 25.4 cm diameter primary mirror. The light source was a star simu lator with measurements made with an AO (11,000*K) star and a GO (6,000°K) star to examine any color effects. The source was chopped at 600 hertz, so that a slowly rotating table could be used and still stay In the star mapper frequency bandwidth (B e e Appendix C). The elevation angle of the star mapper, relative to the collimated beam, w b b varied by use of sine bars. An optical encoder, with a resolution of 2 arc seconds, was used to measure the star mapper azimuth angle. The small hole located In the center of the reticle, shown In Figure 52, was used to define 0° azimuth and 0° elevation. The parameter y shown in this figure Is used to illustrate the azimuthal angular dimen sion of a grating element of either the vertical or slanted coded slit group. The elevation angle was varied from -2.9* to +2.9*, and the rotary table scanned the field of view of the star mapper through the collimated beam at each elevation setting. An outline of the reticle reconstructed by th is means Is shown in Figure 53. The reconstructed reticle was determined to agree with the theoretical reticle to within 0.01* at all points. Determination of the S lit Parameter Angles # The angular relationship between a slit plane, defined by the a a a i10 - plane,and the i1Q - kg plane of Figure 3, has been defined as the angle 6. A summary of the geometry Involved In defining the relationship of this angle to a star sighting vector is given in Chapter 2. For the tests discussed in this appendix, the star mapper was mounted on the rotary table so that, as nearly as possible, the 258 r r irvmM 9.25 ^ • I IM Ml I Figure 52. Schematic of Theoretical Reticle. ,1204 ♦—.1207® 3433' Figure 53. Reconstructed Reticle Outline. 260 vertical slit group was perpendicular to the azimuthal plane of the table. The nonvertlcalneas of the vertical allts to the table azimuthal plane la accounted for In the computation of 0. Initially the star mapper and table were positioned auch that the collimated beam was at center of the alinement hole (0*, 0°). With the table locked at an elevation of 0°, the table was rotated in the azimuthal plane until the collimator bean was coincident with the leading edge of the vertical slit group at a point defined aa VQ. The azimuthal angle of this rotation, defined as py , was recorded. The table was then returned to the alinement point (0°, 0°) and rotated in the opposite direction until the collimator beam was coincident with the leading edge of the slanted slit as so and the azimuthal separation between the alinement point and the leading edge of the slanted s lit, defined as p8 , was recorded also. In Figure 54, ON defines the axis o about which azimuth was measured and F is the alinement point which defines 0° elevation and 0° azimuth. When the table was returned to the (0°, 0°) position and elevated to an angle £, the elevation axis was then positioned at P. with NP defined by an arc of length (90° - O as shown in Figure 55. The azimuthal angle pg^ was measured by the rotation about ON from to V The spherical angle s^ Nsq is denoted by X where The angle 0^ defines the angle of nonvertlcalness of the slanted s lit group with the azimuthal plane of the table. 261 N ao Figure 54. Geometry of Reticle Measurements for the Determination of 3^. 262 From the law of sines and coslneB for spherical triangles [ref. 60], tanX cot s s_ Bln 0. ------— ------(F.2) 1 cos,. and cos 0.1 ■ tan? cot a oa ? (F.3) Dividing equation F.2 by equation F.3 gives tan(p r - p ) ™ This same process was repeated for the vertical slit group to obtain a corresponding angle 0j between Its leading edge and the plane perpen dicular to the azimuthal plane of the table at point on the leading edge of the vertical slit group. The resulting equation, similar to that for 0^, defines 0g tan(p - - p ) t “ *2 ■ — — • ( r -5) The angle between the slanted silt plane and the 1^ - kg plane of Figure 3 is defined by the angle 03 as 0a - 6 i + 02 (F. 6) * * and the angle between the vertical slit plane and the I^q - kg plane Is defined as 0v - 0 • (F.7) 263 In each case the slit plane is defined as the I^ q - plane of Figure 3. Measurements were also made to determine the azimuthal separation between the leading and trailing edges of the vertical and slanted slit groups with each basic slit denoted by p as shown in Figure 53. The azimuthal angular relationship of the vertical slit plane to the optical axis is defined as Yv " PVo “ 9*25p (F.8) The azimuthal angular relationship between the slanted slit plane and the optical axis is defined as (F.9) The additional angular displacement of 0.25y was used to allow time for decision after star transits across the coded slits. The preflight measurements indicated a different value of p for the vertical and slanted slit groups. The measured values of p for the vertical and slanted slit groups were used in equations F.8 and F.9t respectively. It is interesting to note that the angle can be directly related to the mlsallnement angle Zy which is defined in Chapter 2 as a positive rotation about the ig axis and is representative of a rotation of the reticle about the star mapper optical axis. Assume that the coordinates of the reticle in the optical image plane are as Illustrated in Figure 55 where the coordinates (x,y) define an image location within the unrotated reticle bounds. Let an image be located at the point (x<;»yg) in the case of the nonrotated reticle. Then the coordinates for this 264 R eticle O utline (xs ,ys ) Figure 55. Reticle Rotation Due To Miaallnement Angle, ■ point In the x,y system, which define the case for a rotated reticle, w ill be x \ /cos ^ -sin \ / S) (F.10) 7J \sm ei cos ) yys, or x « cos e. x — s in e. y s i s 1 7S y - s in c. x + cos e, y *S 1 S 1 Dividing x by y gives s s 265 — ■» tan 3. (F. 11) Xj .in Cl ys ' CM ci (F.12) x_ sin e. 1 + — i y s cob Cl or / -1 xs -1 8ln cl \ tan B. « tan f tan ---- tan ) • (F.13) 1 I ys c o b J Then 8 1 ■ 8b - C1 or Ba ■ (F.14) which agrees with equation F.6o If the table top used In the measure ments could be assumed to be perpendicular to the principal spin axis of the spacecraft, the measured value of Bg could be used as an Initial estimate for e^. APPENDIX G RELATIONSHIP OF ATTITUDE DETERMINATION ERRORS TO ERRORS IN THE SYSTEM PARAMETER ESTIMATE The set of nine parameters, to be estimated through application of measured transit times of identified stars to a least squares processing routine, has been defined in equation 3.8 of Chapter 3 as x - (e, v»G, 4, For the case of zero torques and negligible asymmetry, all the para meters of this equation may be considered to have a well-defined constant value established at some specified initial value of time. The estimation procedure of this study uses batch processing of transit time data, along with catalog values of right ascension and declination for the known or identified stars. The estimates of the parameters of X represent values of these parameters expressed at some Instant of time such as the beginning of the data fitting or batch processing time period. Clearly this problem may be considered an initial value problem. The next step in the use of the resulting estimates of X is to apply these parameters to the equations of motion, and at specified tlmeB to determine the attitude (in terms of right ascension and declination) of selected axes. Since X will contain errors, the attitude computations will also contain errors. The purpose 266 267 of this appendix Is to Identify the nature of the attitude determination errors as a function of the errors In X. This functional relationship should provide a means for determining the sensitivity of the attitude determination errors to errors In the elements of X. The axes chosen for this Investigation are the ig and kg axes defined In Chapter 2. Since we are only concerned with the ig and kg axes of the star mapper reference frame, the angles y , 3 ,, Y_, and 3 of Chapter 2 will V V 0 8 not be included in this analysis. For convenience let a and b - right ascension and declination of £g, respectively, and A c and d - right ascension and declination of kg, respectively . A * The right ascension and declination of lg and kg are illustrated in Figure 56. 1 -c 1 1 Figure 56. Right Ascension and Declination of i g and k g . 268 The relationship of these two reference frames may be written as I t 1 i ------CO CO H* I-* ------D E H ^8 ^1 (G .l) _ f c 8 . ______1 1 with D, E, and U as defined In Chapter 3. The coordinates of lg and kg e e a may be expressed In the (1^, k^) reference frame as cos b cos a cos b sin a (G.2) sin b 'c o s d cos c and -cos d sin c (G.3) sin d Right Ascension and Declination Error Sensitivity for the lg Axis From equations G.l and G.2 we have A tan a ■ (G.4) B 18 Dx E 1 2 where Is the first row of D and H and H indicate the first and second columns of H, respectively. From these same two equations we have sin b - lg • kx - Dx E r (G.5) JL * K * JL where < k < 2 269 From equation G.4 we have BAA - AAB Aa - — =------=- , (G.6) A + B I t can be shown th at 2 2 2 k* + B* - c o b b then cos bAa ■ . (G.7) COS D The terms AA and AB are expressable as AA - (Dx E Hg) AO + (Dx E H2) A4 + (D^ E^ H2) (Aif>o + tA^) + (Dx E^ H2) (A$o + tA$) + (Dj^ E0 H2) A0 + (Dl e l E H2) Aex + (Dle2 E H2) Ae2 (G.8) AB - (Dx E Hq ) A9 + E h J) A# + (Dx E^ H1) (A\|io + tAtfi) + (Djl E^ H1) (A*o + tA*) + (Dx E0 H 1) Afl + The use of the parameter subscripts on the matrices D, E, and H denotes partial derivatives of those matrices with respect to the subscript parameter. A digital computer program was written to aolve equation G.7 with the use of equations G.4, G.8 and G.9. However, the expressions for cos bAa and Ab will be examined here to illustrate the more critical parameters. The expression for cos bAa is a lengthy one and will be shown here in separate parts corresponding to each element of AX. For simplicity, the parameters e^ and e2 will be set at zero aB they are fixed vehicle parameters and are not dependent on launch time or vehicle dynamics. The components for cos bAa are 270 B(D1 E H2) A0 - A(DX E H*) AO (cos bAa)^g - cos b E h J) A* - A(D2 B(DX E H*)9 A* (cos bA a)^ cos b B(DX H2) (A^o + tA«) - A(D2 (cos bAa)^ ------— j:------(G.10) cos b B(D1 H2) (A*o + tA$) - A(DX E* (cos bAa)., cos b b(d 2 q E0 H2)A0 - A(DX E H1)A0 (cos bA a).ft * ------r ' A0 cos b After expansion and collection of terms, equation G.10 may be written as listed In Table 19. An expression for the declination error, Ab, is found from equation G.5 4b - sir A(DiE h3) As for the right ascension error term, the misallnement terms c^ and e2 will be neglected. Since H3 - (0 sin 0 cos 0)T , D1 E H3 • E12 sin 0 + El3 cos 0 (G.13) ■ sin 0(cos ^ sin $ + cos 6 cos $ sin \|i) + coe 0 sin 4* eln 6 . (G.14) TABLE 19 ELEMENTS OF THE SIGHT ASCENSION ERROR TERM, cos bAa (I cos a oaajA0 sAfll =» =« -(cosfr co84> - cosfl slnj> sinifr) [sln8(cos^ cos sin$ b + cos8 cosfl simji) + cos9 sltnfr sln8]AQ 2 2i (cos bAa) « f(cosfr coa$ - cosB sinA Blnfr) + [cos6 (co3 slniji) - slnO slnt[> slnBl lAft A# * \ cos b (cos bAa) * (cos9 cosB - sln8 sing cos$)_ (A»ft + _tAf), A4> cos b 2 2 2 * [cos9(cos ij> + cos 6 sin ifi) - sin9 siniji sin6 (cosi|» sin$ + cos9 slnif* cos$)] (A$q + tA 2 (cos bA ) - t-alnji cos4>(cos9 slnB 4- sln8 cos$ cos9) + slnQ ain$ sin it>] A9 A9 * cos b 272 Substitution of equation G.14 Into equation G.12 results In the components of declination error shown in Table 20. TABLE 20 ELEMENTS OF THE DECLINATION ERROR TERM, Ab (Ab) ■ ^coaQ(coai|> + cos6 cosA slnA) - ainO sinA sin0 j v fsinOC-sintli a in A + cos0 cosA cosA) + cosG cosA sinO l,*, ttb >4* ■ L 4---- * ---- * ------S T b ---- J«*o+t»*> sinG(cosA cosA - cosG ain (AM « f-sinQ Bin9 cosj a in A + cosO slnA cosO*] .. A0 L c°s h J Examination of the sensitivity coefficients of Tables 19 and 20 reveals some interesting points. The coefficients are all trigonometric functions. The numerators of these coefficients are linear combinations of sines and cosines and under the very worst condition could not exceed a value of 3 or 4. The denominator of these coefficients in all cases is cos b. The value of b has been defined to have the limits of + 90 degrees. However, to hold the error coefficients to reasonable values, simulation runs and flights will be controlled to limit b to values within the bounds of + 60 degrees. The major contributor to 273 both the right ascension and declination error terms is the elements containing tA$ and tAty* Clearly, with A$ and/or Aty having non-zero values, these elements could produce increasingly larger errors if t were permitted to grow without bounds. The seriousness of these elements regarding extrapolation of attitude determination outside the data fitting period w ill be examined. Right Ascension and Declination Error Sensitivity for the £g Axis Using equations G.l and G.3, we have k„ * j , D,EH2 _ tan c - J ------J, . -J . § (G.15) k8 *ii D3 EH1 ,S and s in d ■ kg * k^ ■ Dg E H 8 . (G.16) In a manner similar to that used to obtain equation G.7 we have cos dCSAR - RAS) SAR - RAS cos dAc ------* ------5------— ' .— . (G.17) S + R2 c08 d Also from equation G.10, ^ - ^Td a(d3 e h3) * (G‘18) q The terms AR, AS and A(Dg E H ) are expressable as AR - (D3el E H2)Ae1 + (D3e2 E H2)Ae2 + (D3 EQ H2)A0 + (D3 E^ H2) (A*q + tA*) + (D 3 E^ H2) (At|>o + tAi) + (D3 E H2)A0 + (D3 E hJ)A# (G.19) 274 AS - (D3el E H^Aej^ + (D3e2 E H^ACj + (D3 Efl H^AO + + (D3 E Hq)A0 + A(D3 E H3) - (D3c1E U3)Ae1 + (D3e2 E H3)Ae2 + (D3 E0 H3)A0 + (D3 E+ H3) (A*o + tA}) + (D3 E^ H3) (Ai|>o + tA}) + (D3 E H3)A0 + (D3 E H3)A* . (G.21) Using the expressions In equations G.19t G.20 and G.21, a computer program was written by Mr. D. E. Hinton o£ the Flight Instrumentation Division to solve for the right ascension ~nd declination error terms of equations G.17 and G.18. The inputs required for the program are » the true values of the nine parameters, e^, e2, 0, 4>, }-fl, 9 and their respective errors, Ae^, Ac2, A0, A$, A$o, A^0,A A^, and A6. In addition, the times of error computations or time Increment must be specified as well as the total time limit (I.e., within the bounds of a least squares data fitting period or outside these bounds In an attempt to determine error growth during extrapola tion) . Selection of the AX values was based ona preliminary least squares processing of simulated transit time data plus time measurement errors having a standard deviation of approximately 20 microseconds. * The AX values were the result of comparing the true X values used in the simulation program (see Chapter 6) with the value of X obtained from the least squares processing. The extrapolation period considered in the computer studies covered approximately one precession period beyond the least squares data fitting period. The results Indicated that errors in attitude 275 determination grev as a function of time with the hounds approximated by Aw^t. This effect can be Illustrated through an examination of the error sensitivity terms of Tables 19 and 20. Considering the two elements (Ab)^ and (Ab)^ of Table 20, and assuming a small angle 8, the most significant terms constituting the sum of these two elements are (Ab)A^ + (Ab)^ » CQ^ -b~ sin© cos ($ + ^)[(A$o +A^q) + t(A$ + A$0 ] (G.22) The portion of the error sensitivity term of equation G.22 which Is time dependent is expressed here as Ab^g(t), where ib18(t) * 0lnB + ♦> (A* + A*> t . <0.23) Now for small 8 (A$ + A^i) « Au^ . Then 4„18 S ettin g C'~co 'b at *ts niaximum value and assuming some constant value for sin© gives Abig(t) « kAu^t . (G.25) From the simulation studies of Chapter 6, values of Aw^ were seen to be of the order of 0.0001 degree per second. Then the product Aw^t becomes approximately 0.002 degree for a t of 20 seconds, which is approximately one precession period. Extrapolation of an additional precession period causes this term to grow to 0.004 degree. 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