## S E H Coplanar

'CONDITION FOR MAXIMUM 1 IF THE ANGULAR RADIUS OF THE EARTH (PI IS CONSTANT. 'CONDITION FOR MAXIMUM 1 IF THE ANGULAR RADIUS OF THE EARTH (PI IS CONSTANT. missions, it is necessary to measure as many existing systematic errors as possible. Although the existence of systematic errors can usually be revealed from the behavior of the single-frame attitude solutions, it is our experience that making a quantitative determination of the individual systematic errors solely from this behavior is almost...

## V

Spectral Distribution of Thermal Emission From the Earth Over Midlatitude Oceans. (Adapted from Lyle 1971 .) Fig. 4-8. Spectral Distribution of Thermal Emission From the Earth Over Midlatitude Oceans. (Adapted from Lyle 1971 .) For attitude work, we would like to use a spectral region for which the Earth has a uniform intensity. The considerable fluctuations in the 11- to 14-jim window can be seen in Fig. 4-7. Similarly, the H20 band intensity near 7jtm depends on the strongly varying...

## Info

Kinematic equations of motion for 511 512 Norm of 759 Representation of attitude by 414 558-559, 564-566 Quick look displays 305 QUOMAC (See Quarter Orbit Magnetic Attitude Control) Radiance profiles, of Earth in infrared 95 Radiation balance, for the Earth 82 Radiation pressure, torque due to 570-573 Radiometric balance Earth sensor 180 Radius of gyration, of gas jet propellant 650 RAE (Radio Astronomy Explorer) Application of block averaging to Attitude software structure 698 Data from...

## FtyAyH

An implicit method can be used directly, yielding Such methods for linear differential equations are known as correct-only methods, for reasons that will be apparent shortly. If we are integrating a system of equations, A is a matrix and Eq. (17-13) requires evaluation of a matrix inverse. (The 1 in the first bracket becomes the identity matrix.) * An alternative procedure is to utilize the last k values of yK and that have been evaluated, regardless of step size changes, rather than requiring...

## Main Body

Spring Libration Damper for a Gravity-Gradient Stabilized Spacecraft Magnetically Anchored Eddy Current Damper. A damper such as that used by GEOS-3 consists of two concentric spheres which can move relative to each other and are separated by silicone oil to provide viscous damping. The inner sphere is attached to a magnet which aligns or anchors itself to the Earth's magnetic field vector. The outer sphere is attached to the spacecraft's boom and is made of pyrolitic graphite for...

## Subsatellite Point

SEASAT Scanwheel Configuration. (Compare with Fig. 4-3.) attitude control properties are described in Section 18.3. Attitude accuracy and reliability is increased by dual scanner systems in particular, and attitude accuracy about the velocity vector, or roll axis, is improved by using the difference between the two observed duty cycles, which is less sensitive to attitude variations than the individual scanner duty cycle measurements. Slit Sensors. Slit sensors of a much different...

## Ii

Lift-Off of a Delta 2914 Launch Vehicle From the Eastern Test Range ai Cape Canaveral, Florida. (Conditions for the launch of the Synchronous Meteorological Satellite shown here were essentially identical with those for the CfS launch which occurred after dark.) stationary over the equator at 114 West longitude. This location permitted television transmissions to remote regions of both Canada and Alaska. The mass of the CTS spacecraft was 676 kg at lift-off, of which approximately 340 kg was in...

## Celestial Latitude Galactic Latitude

THE PRESENT SYSTEM OF GALACTIC COORDINATES, ADOPTED IN 1flS8. DEFINES THE NORTH GALACTIC POLE AS BE ING AT o 182.28 . > +27. AND THE GALACTIC CENTER AS AT a 2SSj5 . A - -2& 9l7a. THE PRESENT SYSTEM OF GALACTIC COORDINATES, ADOPTED IN 1flS8. DEFINES THE NORTH GALACTIC POLE AS BE ING AT o 182.28 . > +27. AND THE GALACTIC CENTER AS AT a 2SSj5 . A - -2& 9l7a. *In addition to the mean motion due to precession, the Earth's true spin axis wobbles with an amplitude of 92 arc seconds (0.0026...

## Star

Simplified Diagram of a V Slit Star Sensor detector. This portion is known as the instantaneous field of view (IFOV). The image definition device may be either a reticle consisting of one or more transparent slits etched on an otherwise opaque plate, or an image dissector tube in which the IFOV electronically scans the FOV. The detector transforms the optical signal (i.e., whatever light is not blocked by the image definition device) into an electrical signal. The most frequently...

## Time

General Thrust Profile for Gas Jet Control Systems electrically before propellant is fed in. As with pressure dependence of the force, the time dependence is sometimes available directly from the calibration data in a suitable form otherwise, the initial thrust and the firing time required to reach full thrust can be used to create a simple piecewise linear model of the increase in force. The response of the spacecraft to control torques is proportional to the time integral of the...

## I

Provides the angular error about both the pitch and roll axes. For the EW scanner, pitch is measured using a binary up-down counter to accumulate encoder pulses from the scan mirror. Encoder pulses are counted during the time an Earth radiance signal is present with the direction of count reversed when the scan crosses the spacecraft pitch axis. When the scan is centered, the up count and the down count are equal and a zero count or null pointing angle is obtained. Roll is measured by comparing...

## In Igrf Worked Exercise By Magnetic Scalar Potential In Geomagnetism

Maximum Deviations of Approximate Models From the Earth's Magnetic Field (From Harris and Lyle 19690 plus secondary currents or whirlpools near the core-mantle boundary which produce local dipoles. These secondary dipoles are then superimposed to produce the observed multipole nature of the field, as well as local anomalies, which are large surface areas where the magnetic field deviates appreciably from the dipole field. The creation and decay of the whirlpools may cause the secular drift....

## Oct 241741

*MSAN DAILY MOTION 18 THE MEAN CHANGE IN TRUE ANOMALY AS VIEWED FROM THE SUN hxe LONGITUDE OF PERIHELION.ST. IS MEASURED FROM THE VERNAL EQUINOX THAT IS, w il + FROM THE VERNAL EQUINOX EASTWARD TO THE ASCENDING NODE AND w IS THE ARGUMENT OF NODE ALONG THE ORBIT IN THE DIRECTION OF THE PLANETS MOTION TO PERIHELION. 4AT EPOCH SEPTEMBER 23.0.1960. w. WHERE a 18 MEASURED ALONG THE ECLIPTIC PERIHELION MEASURED FROM THE ASCENDING

## Position Plus Rate Controller

Position-Phis-Rate Pitch Control Block Diagram. See text for explanation. Comparing Eq. (18-9) with the second-order equation of a mass-spring-damper system (see, for example, Melsa and Shultz 19690, we define the natural frequency, < i> , and the damping ratio, p, of our system as UNoy s2+2puns+ We now discuss the response of this system when the input disturbance torque is a step function. Because the Laplace transform of a step function of magnitude N0 is N0 s (see Appendix...

## Sun Angle In Degrees

Plot of the Output From Representative Photocells Versus Sun Angle for Adcole Digital Sun Sensors The fine bits in Fig. 6-14 are used by an interpolation circuit to provide increased resolution. Straightforward addition of encoded rows to the pattern is not possible because th 0.53-deg angular diameter of the Sun from near the Earth would blur the output from adjacent bits. This effectively limits Gray 'code transitions to a 1 2-deg LSB. By combining the output of 2 or 3 offset LSB...

## Magnetometer

_ REFERENCE (redundant-optional) UNIT IIRU) Fig. 22-5. Attitude Control System Module Within the MMS Spacecraft onboard processor and the ground software Werking, 1976 , Definitive attitude will be determined by the onboard processor. For SMM, the onboard computer will spend approximately 30 of its time in attitude-related activities. Ground processing will be used for early mission support (stabilization and attitude acquisition), for calibration or bias determination (to determine sensor...

## Spin Stabilized Spacecraft

Our first example of the attitude determination process is the case of the CTS spacecraft in its transfer orbit as described above. Because the purpose of attitude det & piation in the transfer orbit is to support an orbit maneuver and the nozzle of the apogee boost motor is aligned with the spacecraft spin axis, we are interested in the orientation of the spin axis in inertial space. As shown in Fig. 1-5, two types of attitude sensors are available. (Two sensors of each type were used,...

## Path Of Bolometer Axis Due To

Bolometer Offset Model Geometry the rotation angle about the spin axis from the Earth-in to the nadir vector, and L0 is the rotation angle about the spin axis from the nadir vector to the Earth-out. B + (Q 2-L,)+tsl(ww-us) + Qb 360 or 720 (7-55) where is the wheel rate, ws is the body spin rate, and < PS is the phase of the bolometer offset. For a fixed bolometer position, the four unknowns in these equations are t), L L0, and B. Equations (7-54) and (7-55) determine B and L0 in...

## ASEpcApAy ususfMWsusfs

E is the unit nadir vector. It remains to express and A 5 in terms of their arguments. For the case of the Sun-to-Earth-in model, < > , and is again calculated from Eq. (7-57) or Eq. (7-59), in which p and y are replaced everywhere by pc + Ap and yN + by, respectively. A s is calculated by applying Napier's rules for right spherical triangles to the lower triangle associated with the Sun sensor in Fig. 7-16, yielding sin A s - tan(90 -0)tan 9O -(90 - ts) Here,...

## Magnetic

'NONSPINNING EARTH SENSOR ASSEMBLY (SIC SECTION 6.7) PROPOSED SPACECRAFT. REGISTERED TRADEMARK OP ITNACO CORPORATION. (SEE SECTION BJ1 4ACR0NVM POR WHEEL CONTROL (SEE SEASAT DISCUSSION IN TEXT) 'NONSPINNING EARTH SENSOR ASSEMBLY (SIC SECTION 6.7) PROPOSED SPACECRAFT. REGISTERED TRADEMARK OP ITNACO CORPORATION. (SEE SECTION BJ1 4ACR0NVM POR WHEEL CONTROL (SEE SEASAT DISCUSSION IN TEXT) * Direct yaw control can be achieved using a gyroscope, as in inertial guidance systems using either onboard...

## Azimuth Slit Window

6.43 BBRC CT-401 Fixed-Head Star Tracker As an example of the fixed-nead star tracker, we will describe the CT-401, manufactured by BBRC and shown in Fig. 6-42. The CT-4Q1 has flown on the SAS-3 and HEAO-1 missions and is scheduled for HEAO-C and MAGSAT. Schematically, the tracker is similar to the instrument shown in Fig. 6-39. The specified accuracy of the tracker over its 8- by 8-deg field of view is 3 arc-minutes without calibration, or 10 arc-seconds after applying corrections for...

## Mounting Bracket

BBRC CT-401 Star Tracker Search and Track Mode Patterns The initiate search command can be used to vary the style of operation. If the tracker is frequently commanded to resume search, it operates more like a star mapper than a tracker. If it is commanded only infrequently, it operates more like a tracker. A bright object sensor and shutter mechanism protect the image dissector from excess energy from the Sun, the Moon, or the lit Earth. The sensor will close the tracker's shutter...

## Ek I 2 Xp Xft i 2 [px

Thus, both the angular momentum and the kinetic energy can be expressed in terms of and < o. Some authors (e.g., Whittaker 1937 ) define the negatives of the off-diagonal elements as products of inertia but other authors (e.g., Goldstein 1950 and Kibble 1966 ) define the elements IJk, without the minus sign, as the products of inertia. Still other authors (e.g., Thomson 1963 and Kaplan 1976 ) define the products of inertia as Whittaker does, but denote them by ljk, so that the off-diagonal...

## Astronomy For Ha

The 0-deg longitude line is referred to as the Greenwich meridian because it is defined by the former site of the Royal Greenwich Observatory. Greenwich Mean Time, GMT, is the mean solar time at 0 deg longitude that is, GMT is the HA of the mean Sun observed at Greenwich (called the GHA) in hours plus 12 hours, modulo 24. Greenwich Mean Time is also called Universal Time, UT, and, in spaceflight operations, Zulu, or Z. Uncorrected UT or UTO (read UT Zero) is...

## Mathematical Models Of Attitude Hardware

- V-Slii Sensors, Digital Sensors 72 Horizon Sensor Models Horizon Sensor Geometry, Nadir Vector Projection Model for, Body-Mounted Sensor, Central Body Width Model, Split Angle Model for Wheel-Horizon Scanner, Biases 13 Sun Sensor Horizon Sensor Rotation Angle Models 7.4 Modeling Sensor Electronics Theory, Example 1R Horizon Sensor Calibration of Vector Magnetometers, Magnetometer Biases Star Scanner Measurements, Image Dissector Tube Star Measurements, Modeling Sensor Intensity Response 7.7...

## N

Choice of the range, N, and weights, ak, permits the selective attenuation of high-frequency components in the data and significantly alters the statistical characteristics of the data.* Removal of high-frequency noise highlights the actual frequency characteristics of the data. The digital filter, Eq. (9-1), is derived from analog filters used for electronic signal processing. Curve fitting is a technique which assumes a functional form for the data over a time interval and computes a set of...

## T0

That with the perturbation, the position errors are almost entirely in track without the perturbation, the out-of-plane error becomes significant, especially for cases 1 and 3. A second study was performed by Legg and Hotovy 1977 for an orbit similar in size and shape to that of case 1 (e 0.001, a 6928 km) but with an inclinatipn of 97.8 deg. This study showed the major perturbations to be due to J2, but that implementing only the secular component resulted in a significant degradation in...

## Effect Of Loss Of Two Most Critical Sensors

VERY CRITICAL ALL SUN OR EARTH OBSERVATIONS LOST SERIOUS AU. SUN ANGLE AND M1DSCAN NADIR ANGLE MEASUREMENTS LOST SUN-EARTH AND EARTH-WIDTH ANGLES 87 COVERED Software Standardization. The need for software standardization comes from increased data volume and the demand for both reduced costs and greater accuracies. By spreading the costs over multiple missions, sophisticated, expensive hardware and ground or onboard processing tools can be developed. In addition, complex systems become more...

## Iffict At Ustir

Earth Width Data for the SMS-2 Spacecraft Illustrating the Pagoda Effect 1. Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions, NBS Applied Mathematics Series 55, 1964. 2. Budurka, W. J. Data Smoothing and Differentiation in Program TRIAD, IBM Report No. TM-67-27, Dec., 1967. 3. Chen, L. C., and J. R. Wertz, Analysis of SMS-2 Attitude Sensor Behavior Including OABIAS Results, Comp. Sc. Corp., CSC TM-76 6003, April 1975. 4. Coriell, K., Geodynamics Experimental Ocean...

## List Of Contributing Authors

AH the authors are members of the technical staff in the Attitude Systems Operation, System Sciences Division, Computer Sciences Corporation. Sections written by each author are in brackets. John Aietto B.S. (Astronomy), Villanova University 5.2, Appendix G Jatvaid Bashir Ph.D. (Aerospace Engineering), M.S. (Electrical Engineering), University of Colorado B.S. (Electrical Engineering), Karachi University, Pakistan 18.1 Robert M. Beard M.S. (Mathematics), B.S. (Physics), Auburn University 16.3...

## H2 z h2 hy h jxy y hz h

+ If X, Y, and Z are principal body axes, then Eq. (17-41) reduces to < N cc > 0--p where hx, hy, and hz are the components of the orbit normal unit vector along the principal axes. From Eq. (17-42), we see that (1) if any principal axis is parallel to orbit normal, the secular gravity-gradient torque is zero and (2) if a principal axis is in the orbit plane, the secular gravity-gradient torque will be along that axis. The secular gravity-gradient torque for a spin-stabilized satellite can...

## Attitude Estimate

Expanded View of Fig. 10-2 Showing Intersecting Attitude Loci for Measurements 0 and ij perpendicular to the loci. Then the measurement density, dm, is the two-dimensional gradient of m on the celestial sphere for a fixed position of the reference vectors (i.e Sun vector, nadir vector, etc.). That is, afm Vm for Fixed reference vectors If we let m, and m2 be the limits of uncertainty in measurement m, then corresponds to the width, Ai.m, of the attitude uncertainty band on the...

## Pulse Counts Sun Angles Frame Times Son Rates Earth Times

Data Flow During Maneuver Monitoring for CTS. The control monitor receives attitude solutions from the deterministic attitude subsystem and the rest of the data from the data adjustment subsystem. Fig. 21-4. Data Flow During Maneuver Monitoring for CTS. The control monitor receives attitude solutions from the deterministic attitude subsystem and the rest of the data from the data adjustment subsystem. and other parameters for the scheduled maneuver. During the actual maneuver the...

## Estimated True Spin Period

Attitude estimate is provided by extrapolating an initial attitude using a model of the spacecraft attitude motion. For example, after triplets of SAS-2 N-slit star scanner observations have been grouped together using the initial spin rate, a better spin rate estimate is obtained from the time differences between the slit 1 and slit 3 crossings within the same triplet This improved spin rate is then used to calculate observed star unit vectors at some nearby epoch time. This set of observed...

## Attitude Stabilization

18.1 Automatic Feedback Control 18.2 Momentum and Reaction Wheels Momentum Bias Control Systems, Reaction Wheel Systems 18.3 Autonomous Attitude Stabilization Systems Inertially Referenced Spacecraft (HEAO-l), Earth-Referenced Spacecraft 18.4 Nutation and Libration Damping Passive Nutation Damping, Active Nutation Damping, Libration Damping Chapters 18 and 19 describe the various techniques used for attitude control. These techniques may be divided into two categories. Attitude stabilization,...

## [1

Equations (6-8) and (6-9) imply that even harmonics of the primary frequency can occur only in the presence of an external magnetic field. The ratio of the second harmonic to the first is 1 - exp - i2tt( 1 -& H Hd ) l-exp - r(l-A tf ) For and Hc< kHd, then and This means that the second harmonic is 90 deg out of phase with the primary. The sign of the second harmonic gives the sense of AW relative to the core axis and the amplitude is proportional to hH HD. The fluxgate external magnetic...

## Angular Misalignment Of Uotor Case With Respect To Spacecraft Centerline Angular Misalignment Of Principal Inertia Axes

Definition of Misalignments for a Rocket Motor Nominally Aligned With the Spin Axis of a Spinning Spacecraft An analytic model for the spacecraft motion during the engine firing, including the above six misalignments, can be developed with the following simplifying assumptions 1. Rigid body dynamics are applicable. 2. The engine is a solid fuel motor and the fuel burns symmetrically about the motor case centerline. 3. The total spacecraft mass, moments of inertia, and the location of...

## Introduction

12 Representative Examples of Attitude Determination and Control Spin Stabilized Spacecraft, Three-Axis Stabilized Spacecraft, Attitude Maneuver Using Gas Jets 13 Methods of Attitude Determination and Control 1.4 Time Measurements The attitude of a spacecraft is its orientation in space. This book is concerned with all aspects of spacecraft attitude how it is determined, how it is controlled, and how its future motion is predicted. We describe simple procedures for estimating the attitude and...

## Status Executive And Buffer Stored Command Processor

SUBSYSTEM FUNCTIONS ACS POWER THERMAL OBSERVATORY SAFETY OBSERVATORY OPERATIONS MISSION UNIQUE FUNCTIONS Fig. 6-51. NSSC-I Memory Layout Tor the MMS Spacecraft The MMS attitude control system core requirements are 10 K. to 16 K. words of program and data storage 10 to 1000 stars at 4 words per star and 100 to 3000 words for 72 hours of ephemeris data NSSC-II. This computer is a microprogrammed general-purpose computer that is compatible with the standard instruction set of IBM S 360...

## Angular I

G., editor, Proceedings of the Magnetic Workshop, March 30-April 1, 1965, J PL Tech Memo 32-316, 1965. 4. Bauer, H. F., Theory of the Fluid Oscillations in a Circular Cylindrical Ring Tank Partially Filled with Uquid, NASA TN D-557, 1960. 5. -, Fluid Oscillations in the Containers of a Space Vehicle and Their Influence Upon Stability, NASA TR R-187, 1964. 6. Beard, R. M., J. E. Kronenfeld, and E. Areu, Small Astronomy Satellite-2 (SAS-2), Dynamic Attitude Determination System (DYNAD)...

## Rocket Thrust To Lower The Or

Note that direction of thrust is opposite the direction propellant is expelled. Figure 3-14 shows that minimum energy maneuvers between orbits start 180 eg away from the desired end location relative to the central body. If subscripts 1, 2 and H denote the two circular orbits and the Hohmann transfer ellipse, respectively, then, from Fig. 3-14, the semimajor axes are related by * fl i i(ai + fl2) (3-18) from which the period, PH, and the transfer time, T...

## Sun Sensor Models

Two-Axis Sun Sensor Optics Fig. 7-7. Two-Axis Sun Sensor Optics where the FORTRAN function ATAN2 is used in Eq. (7-18) to resolve the quadrant ambiguity. If we substitute Eq. (7-16) into (7-19), and rearrange terms, we get The angles and 9 are the azimuth and coelevation, respectively, of the Sun vector in sensor coordinates which have the positive pole along the sensor boresight and the reference meridian along the + XS axis. The Sun vector may be transformed into spacecraft...

## Thermopile

Lal ARRANGEMENT OF THERMOPILES OF SYMPHONIE STATIC EARTH SENSOR lal ARRANGEMENT OF THERMOPILES OF SYMPHONIE STATIC EARTH SENSOR toi RADIOMETRIC BALANCE HORIZON SENSOR USED FOR ERTS LANDSAT toi RADIOMETRIC BALANCE HORIZON SENSOR USED FOR ERTS LANDSAT Magnetometers are widely used as spacecraft attitude sensors for a variety of reasons they are vector sensors, providing both the direction and magnitude of the magnetic field they are reliable, lightweight, and have low power requirements they...

## Linear Acceleration Sensitivity

Because of its high accuracy and low drift, the rate-integrating gyro is the type most often used in spacecraft attitude sensing. The gimbal is mounted so that its motion is essentially frictionless and without spring restraint. It is usually a sealed cylinder which is immersed or floated in a fluid. The spin axis in the cylinder is generally supported by either gas or ball bearings. Because the viscous damping and spring constants are both small, the steady-state solution to Eq. (6-15)...

## E

Sample Output From Simulator Used for Comparison of Predicted and Observed Data. Plus signs represent observed data solid lines represent predicted data for two distinct attitude solutions. Sample Output From Simulator Used for Comparison of Predicted and Observed Data. Plus signs represent observed data solid lines represent predicted data for two distinct attitude solutions. 1. Fallon, L., D. M. Gottlieb, C. M. Gray, and S. G. Hotovy, Generalized Star Camera Attitude Determination System...

## Summary Of Orbit Properties And Terminology

3.5 Viewing and Lighting Conditions This chapter provides background information and defines the terminology used in orbit analysis and mission planning. Approximate expressions, appropriate for hand calculation, are provided and motivated so far as possible. Many general technical works are widely available in the areas of celestial mechanics and orbit analysis among the more popular are Baker 1967 , Baker and Makemson 1967 , Battin 1964 , Escobal 1965 , Herrick 1971 , Kaplan 1976 , McCuskey...

## O

Greenwich Mean Time is used because 0 deg longitude is defined as going through the site of the former Royal Greenwich Observatory in metropolitan London. Eastern Standard Time in the United States is obtained by subtracting S hours from the Universal Time. Because calendar time is inconvenient for computations, we would like an absolute time that is a continuous count of time units from some arbitrary reference. The time interval between any two events may then be found by simply subtracting...

## Ib

Where R, is the vector from the spacecraft center of mass to the center of pressure of the ith element. The center of pressure is at the intersection of the line of action of the single force which replaces the resultant radiation force and the plane passing through the center of mass of the spacecraft perpendicular to the line of action. The location of the center of pressure, r , relative to the centroid of the geometrical sphere is given by Solar radiation torques are reduced by the shadows...

## Attitude Geometry

2.1 The Spacecraft-Centered Celestial Sphere Properties of Coordinate Systems on the Celestial Sphere, Spacecraft-Centered Coordinate Systems, Nonspacecraft-Centered Coordinate Systems, Parallax 23 Elementary Spherical Geometry This chapter introduces the idea of attitude determination and control as a geometrical problem on the two-dimensional celestial sphere, describes the most common attitude coordinate systems, and summarizes geometry on the celestial sphere. 2.1 The Spacecraft-Centered...

## Horizon Plane

Meridian Cross Section of the Earth Showing the Horizon Spheroid and the Horizon Bane. (In this Figure, the observer is in yz plane.) The normal to the horizon plane depends only on the angular position of the observer. Thus, as the observer moves along a fixed nadir* direction, he sees a set of parallel horizon planes. As shown in Fig. 4-17, when the observer is on the Earth's surface (point A), the horizon plane is just the tangent plane at that point. As the observer moves to a...

## Construction Of Global Geometry Plots

Global geometry plots, as used throughout this book, are convenient for presenting results and for original work involving geometrical analysis on the celestial sphere. The main advantage of this type of plot is that the orientation of points on the surface is completely unambiguous, unlike projective drawings of vectors between three orthogonal axes. This appendix describes procedures for manually constructing these plots. Methods for obtaining computer-generated plots are described in Section...

## Time Interval Iseci

Model of Attitude Determination Uncertainty Due to Gyro Noise Effects Using Parameters Specified for the High Energy Astronomy Observatory-1 Mission. (Adapted from McElroy, 11975J.) If active spacecraft control is to be modeled in a simulation system, mathematical models of the control system are needed. Specifically, models of the reaction wheel torque and friction characteristics are needed to model reaction wheel control systems. As discussed in Section 6.6, reaction wheel...

## Spherical Harmonics

Laplace's Equation, V2U 0, can be written in the spherical coordinate system of Section 2.3 as If a trial substitution of U(r,9,< p) R(r)Y(9,< l> ) is made, the following equations are obtained through a separation of variables d2R(r) d R(r) r2- y-+2r- --n(n+l)R(r) 0 (G-2) where n(n + 1) has been chosen as the separation constant. Solutions to Eq. (G-2) are of the form Thus, solutions to Laplace's Equation (Eq. (G-l)) are of the form U Arn + Br-< + ) Y(9,4> ), n 0,l,2, , (G-5) These...

## Unlimited

TVPICAL TORQUE ABOUT THE WHEEL AXIS 0 1 N-M B S (H) 7.9 IH) IB J IS), 19.4 (M) IB.3 IS) f - ENVIRONMENT MODEL H - HARDWARE. S - STABILIZATION. U - MANEUVERS f - ENVIRONMENT MODEL H - HARDWARE. S - STABILIZATION. U - MANEUVERS Passive Attitude Control The most common passive control techniques are spin stabilization, in which the entire spacecraft is rotated so that its angular momentum vector remains approximately fixed in inertial space dual-spin...

## 1

0330 0340 0360 0300 0310 0330 0430 M35 0450 0466 Fig. 19-20(a). Observed and Simulated Attitude Data During GEOS-3 Attitude Acquisition. Attitude data immediately after boom extension and wheel acceleration. 2. Basset, D. A., Ground Controlled Conversion of Communications Technology Satellite (CTS) From Spinning to Three-Axis Stabilized Mode, Paper No. A AA 76-1928, AIAA Guidance and Control Conference, San Diego, CA, Aug. 1976. 3. Blaylock, B. T., and R. Berg, International Ultraviolet...

## Measurement Component

Two One-Axis Sun Sensors for Spinning Spacecraft With 180-Deg FOV (Adcole Model 17083.) Fig. 6-12. Two One-Axis Sun Sensors for Spinning Spacecraft With 180-Deg FOV (Adcole Model 17083.) The measurement component generates an output which is a digital representation of the angle between the sunline and the normal to the sensor face when the Sun is in the FOV of the command component, as shown in Fig. 6-13. The measurement component illustrated in Fig. 6-14 is a composite (similar to...

## JgjAI

Where a, is the right ascension of the mean Sun*, S2 is the right ascension of the ascending node of the orbit, i is the orbital inclination, AD is the number of days from the vernal equinox to the time i2 is evaluated, the angular quantities in square brackets are in degrees, and T is in hours. For Sun-synchronous orbits, as - S2 is constant For the orbit to be Sun synchronous, dQ dt must be positive and i must be greater than 90 deg that is, it must be a retrograde orbit. Such orbits are...

## Ring Oscillating About An Axis Passing Through The Rim

Where K is a constant which depends on the shape and size of the poles. Eddy current dampers have the advantage of a variable damping constant, because the strength of the electromagnet can be changed by ground command. The SAS dampers could also be tuned in flight for different spin rates by changing the spring stiffness of the damper and hence its frequency of vibration. Ball-ln-Tube Damper. A ball-in tube damper, shown in Fig. 18-21, consists of a closed, curved tube in which a ball is...

## Aperture Plate

Two-Axis Mask Sun Detector Schmidtbauer, et al., 1973 . The sunline is normal to the aperture plate if the output of all four solar cells is equal. Sun presence detectors are used to protect instrumentation, to activate hardware, and to position the spacecraft or experiments. Ideally, Sun presence detectors provide a step function response that indicates when the Sun is within the FOV of the detector. For example, the shadow bar detector shown in Fig. 6-6 has a steep output slope and,...

## Nl

Here, Br is the radial component (outward positive) of the field, Bt is the coelevation component (South positive), and B is the a imuthal component (East positive). (See Fig. 2-5, Section 2.3.) The magnetic field literature, however, normally refers to three components X, Y, Z, consisting of North, East, and nadir relative to an oblate Earth. These components are obtained from Eq. (H-12) by X( North) - Bgcost-Brsin Y(East) Bq (H-13) Z( Vertical inward positive) 0sin< - B,cost where t A-S<...

## U

As shown in Fig. 19-12(c), after the trajectory intersects the switching line, it will proceed directly toward the origin. For example, an initial state at A will intersect the switching line at B, change directions, and move along the switching line to the origin where the control is set to zero. Similarly, an initial state at C will intersect the switching line at D and proceed to the origin. This case is discussed in more detail by Hsu and Meyer 1968 . In practical applications,...

## Disadvantages

LOW POWER AND WEIGHT.' USUALLT MUST BE KNOWN 'ORSOLAR CELLS AMO EQUIPMENT PROTECTION MAY NOT BE VISIBLE DURING PARTS OF ORBIT ' AROUND LARGE CENTRAL BODY 1 2-DEG ANGULAR DIAMETER (VIEWEDFROM EARTHI LIMITS ACCURACY TO -1 ARC MINUTE ALWAYS AVAILABLE FOR NEARBY SPACECRAFT. BRIGHT LARGELY UNAMBIGUOUS (MAY BE MOON INTERFERENCE) NECESSARY FOR MANY TYPES OF SENSOR AND ANTENNA COVERAGE. ANALYSIS RELATIVELY EASY TYPICALLY REQUIRE S SCAN MOTION TD SENS HORIZON. SENSORS MUST BE...

## Spherical Geometry

A.2 Right and Quadras ta) Spherical Triangles A.3 Obliqbe Spherical Triangles A.4 Differential Spherical Trigonometry A.S Haversines Finding convenient reference material, in- spherical geometry is difficult. This appendix provides a compilation of the most useful equations for spacecraft work. A brief discussion of the basic concepts of spherical geometry is given in Section 23. The references at the end of this appendix contain further discussion and proofs of most of the results presented...

## Fc Swto 2FM

This example shows that the matrix elements of f(M) are not the functions f of the matrix elements of M, in general. Let be a scalar function of the n arguments XltX ,X . We consider-the arguments to be the components of a column vector The n partial derivatives of < j> with respect to the elements of X are the components of the gradient of < f> , denoted by Note that d< j> dX is considered a 1 Xn row matrix. If we eliminate the function < j> from Eq. (C-80). we obtain the...

## M

An application of this theorem to Eq. (12-72a) with implies that, for m large, the errors in xt, that is, Sx are approximately Gaussian with standard deviation In practice, the Central Limit Theorem may give reasonable estimates for m as small as 4 or 5, although in such cases the results should .be verified by other means. The Central Limit Theorem may also be used to compute the variance and distribution of errors in a measurement which is contaminated by many error sources with (presumably)...

## Introduction To Attitude Dynamics And Control

15.3 Introduction to Attitude Control Dynamics is the study of the relationship between motion and the forces affecting motion. The study of the dynamics of objects in interplanetary or interstellar space is called astrodynamics and has two major divisions celestial mechanics and attitude dynamics. Celestial mechanics or orbit dynamics, discussed briefly in Chapter 3, is concerned with the motion of the center of mass of objects in space, whereas attitude dynamics is concerned with the motion...

## Aititude Hardware

Analog Sensors, Sun Presence Detectors, Digital Sensors, Fine Sun Sensors Sensor Components, Horizon Sensor Systems 6J Magnetometers 6.4 Star Sensors Overview of Star Sensor Hardware, BBRC CS-103 V-Slit Star Scanner Jbr OSO-8, BBRC CT-401 Fixed-Head Star Tracker 6J5 Gyroscopes Rate Gyros, Rate-Integrating Gyros, Control Moment Gyros 6.6 Momentum and Reaction Wheels In this chapter we describe representative examples of spacecraft hardware used for both attitude determination and attitude...

## Teo

The conjugate or inverse of q is defined as The quantity, qA, is the real or scalar part of the quaternion and iqt + jq2 + kq3 is the imaginary or vector part. A vector in three-dimensional space, U, having components ,, U2, U3 is expressed in quaternion notation as a quaternion with a scalar part of zero, If the vector q corresponds to the vector part of q (i.e., q iqt + jq2+kq3), then an alternative representation of q is Quaternion multiplication is performed in the same manner as the...

## Q qqm qq q2q2 q3q3 qrfj 23

+j( lifi + + + Ml) + k(qlq'2-q2q + q3q +q q3) (D-7) If q' (q ,q'), then Eq. (D-7) can alternatively be expressed as q -qq' (q - q q', 4q'+ q'A + qX q) The length or norm of q is defined as If a set of four Euler symmetric parameters corresponding to the rigid body rotation defined by the transformation matrix, A (Section 12.1), are the components of the quaternion, q, then q is a representation of the rigid body rotation. If q' corresponds to the rotation matrix A', then the rotation described...

## P

Typical Photodiode Schematic Fig. 6-21. Typical Photodiode Schematic Detectors which respond to the longer wavelengths of infrared blackbody radiation are based on the operating principles of the thermistor, thermocouple, or pyroelectric crystal Barnes Engineering Co., 1976 . A bolometer is a very sensitive resistance thermometer, or thermistor, used to detect infrared radiation. Thermistors consist of fused conglomerates, or sinters, of manganese, cobalt, and nickel oxide formed...

## Preface

The purpose of this book is to summarize the ideas, data, and analytic techniques needed for spacecraft attitude determination and control in a form that is readable to someone with little or no previous background in this specific area. It has been prepared for those who have a physics or engineering background and therefore are familiar with the elementary aspects of Newtonian mechanics, vector algebra, and calculus. Summaries of pertinent facts in other areas are presented without proof....

## I9229i9b 9i94 29I92 9394 99293 92 29i93 9294 2ili3 qxq4 q qj q

The inverse transformation matrix in this case may be obtained by use of Eq. E-4 , or by replacing qt,q2, and q3by -qx, q and - q respectively, in Eq. E-8 and leaving q4 unaltered. The Euler symmetric parameters may be regarded as components of a quaternion, q, defined by where i, j, and k are as defined in Appendix D. The multiplication rule for successive rotations represented by Euler symmetric parameters is given in Appendix D. The Euler symmetric parameters in terms of the 3-1-3 Euler...

## Quaternions

The quaternion representation of rigid body rotations leads to convenient kinematicai expressions involving the Euler symmetric parameters Sections 12.1 and 16.1 . Some important properties of quaternions are summarized in this appendix following the formulation of Hamilton 1866 and Whittaker 1961 , Let the four parameters form the components of the quaternion, where i,j, and k are the hyperimaginary numbers satisfying the conditions

## Reference Axis

Orientation of Two Cosine Detectors To Provide Sun Angle Measurements Over a Wide Angular Range Fig. 6-2. Orientation of Two Cosine Detectors To Provide Sun Angle Measurements Over a Wide Angular Range each sensor the solid line is the summed output each sensor the solid line is the summed output A second analog sensor type uses a bar or mask to shadow a portion of one or more photocells. Different configurations can yield a one-axis sensor Fig. 6-4 or a two-axis sensor Fig. 6-5 with...

## Inter Plane Tar Y Space Asovc Synchronous Altitude

LARGELY INDEPENDENT HIGH CONCENTRATION IN SONS REGIONS OP THE SOLAR SYSTEM NORMALLY NEGLIGIBLE MAY BE IMPORTANT IN SOME 9 ALL REGIONS INTERIOR OP SATURN'S RINGS 'ALTITUDES LISTED ARE ONLY REPRESENTATIVE THE SPECIFIC ALTITUDES AT WHICH VARIOUS TORQUES DOMINATE ARE HtOHLV SPACECRAFT DEPENDENT. three regions where different forces dominate. Close to the Earth, aerodynamic torque will always be the largest. Because this falls off exponentially with distance from the Earth, magnetic and...

## Threeaxis Attitude Determination Methods

12.1 Parameterization of the Attitude 12.2 Three-Axis Attitude Determination Geometric Method, Algebraic Method, q Method Chapter 11 described deterministic procedures for computing the orientation of a single spacecraft axis and estimating the accuracy of this computation. The methods described there may be used either to determine single-axis attitude or the orientation of any single axis on a three-axis stabilized spacecraft. However, when the three-axis attitude of a spacecraft is being...