THE U N I V E R S IT Y OF M I C H I G A N COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report THE ATTITUDE OF THE THERMOSPHERE PROBE M. F.'Carter ORA Prpoje.t 07- 6 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GODDARD SPACE FLIGHT CENTER CONTRACT NO. NAS 5-9113 GREENBELT, MARYLAND administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR April 1968

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TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF SYMBOLS v 1. INTRODUCTION 1 2. MODEL OF MOTION OF THE TP 2 2.1. Analysis of a TP Ejection 2 2.2. Design of the Ejection System 13 2.3. Motion of the TP after Ejection 16 3. ATTITUDE SENSING INSTRUMENTS 17 3.1. Solar Aspect 17 3.2. Lunar Aspect 19 3.3. Velocity Vector Aspect 19 3.4. Earth Normal Aspect 19 4. PARAMETERS OF THE ATTITUDE SOLUTION 20 4.1. Tumble-Plane Coordinates 20 4.2. The Parameter 7 22 4.3. The Parameter 0LS 23 5. SOLUTION FOR ATTITUDE 25 5.1. Solution for OLS 25 5.2. Calculation of Angle of Attack 30 APPENDIX A. MEASUREMENT OF y 32 APPENDIX B. GENERAL OLS ANALYSIS 35 REFERENCES 38

LIST OF FIGURES Figure Page 1. The yo-yo despin module. 3 2. Structure of the nose cone and the ejection system. 4 5. The negator causing the TP to tumble. 6 4. Velocity imparted by the plunger spring as viewed from the center of mass. 7 5. Computer format for the mathematical solution of a typical ejection of a TP. 12 6. Final tumble rate versus negator force for a typical TP. 14 7. Extension of the negator cable versus length of travel of the plunger for a typical TP. 14 8. Separation velocity versus length of travel of the plunger for a typical TP. 15 9. Final spin rate versus initial wrap angle for a typical TP. 15 10. Motion of the TP in free space. 16 11. Measurement made by the solar sensor. 18 12. Tumble-plane coordinate system. 21 13. Solar sensor triggering. 22 14. Sun-velocity coordinate system. 26 15. Aspect solution in the tumble-plane coordinate system. 26 16. Typical plot of y versus 4LS' 29 17. General eLS analysis. 35 iv

LIST OF SYMBOLS -, A A A an intermediate vector in the solution for 1LS equal to V x L A A C L x w f angular deviation of the solar sensor field of view from the solar sensor plane E vector from the center of gravity of the TP to the point where the negator cable meets the TP F a vector fixed with respect to the TP, or a force vector g gravitational acceleration h altitude H vector from center of gravity of the TP to the negator hook I1,I2, I3 elements of inertia tensor of the TP in the TP coordinate system k Boltzmann' s constant K spring constant of plunger L unit angular momentum vector m molecular mass of gas M1 mass of the TP M2 mass of the plunger M3 mass of the burned rocket and nose cone assembly without the TP n ambient particle density a ni particle density inside the gauge volume N earth normal vector R reference vector, or displacement vector S normal speed ratio V cos o/'2kTa/m v

LIST OF SYMBOLS (Continued) SO speed ratio V/42kTa/m S unit vector toward the sun t time tn time of the nth occurrence of an event Ta ambient temperature u,v,w coordinate axes of the TP system V unit velocity vector V total velocity of the TP, IVI Vo velocity of separation of the TP and the rocket Vz vertical velocity of the TP Xmax maximum compression of the plunger spring Xmin minimum compression of the plunger spring A A A x,y,z coordinate axes of a local reference frame A A A X,Y,Z coordinate axes of a local reference frame A A a instantaneous angle of attack, the angle between V and w CRF the angle between R and F amin minimum angle of attack during a tumble A 6 coning angle, the angle between w and the tumble-plane C angular shift from ~+ /2 in the tumble-plane of w(sun pulse) Q* angular parameter used in OLS measurement A A OF angle between F and w A A OLS angle between L and S 0LM angle between L and the vector toward the moon vi

LIST OF SYMBOLS (Concluded) A A 0LV angle between L and V A A QSV angle between S and V A A EXS angle between X and S T vector representing the moment imposed by the negator on the TP A A A A O the ~-spherical coordinate of F in X, Y. Z system A OLF the ~-spherical coordinate of F in the TP system "LO the 0-spherical coordinate of w in the tumble-plane system A ~)LS the ~-spherical coordinate of L in the sun-velocity system On angle measured by the solar sensor at time tn the ~-spherical coordinate of L in the TP system XV the ~-spherical coordinate of V in the tumble-plane system Jo W,2 W3 the components of Q in the TP system Wt tumble rate, the magnitude of the rate of rotation of the TP transverse to its longitudinal axis Q rotation vector of the TP go constant rotational rate of the rocket at the time of ejection vii

1. INTRODUCTION The Thermosphere Probe (TP) is a rocket-borne system which contains various experiments to investigate the upper atmosphere (Spencer, et al., 1965, and Taeusch, et al., 1965). The instruments it carries measure geophysical parameters in the altitude range of 100 to 350 kilometers. The instruments normally included in the TP system are a Langmuir probe to measure electron temperature and density and one or two mass spectrometers to measure either neutral particle density and temperature or positive ion density. In order to accomplish these experimental objectives, the TP employs a cylindrical configuration and a tumbling motion. The TP is a cylinder six inches in diameter and approximately three feet in length. One or two mass spectrometers are mounted so that each has its orifice centered in an end of the TP. The Langmuir probe is mounted in the center section along with an optical aspect sensor and the telemetry antennae. During boost, the TP remains inside the nose cone of a Nike-Tomahawk vehicle. When the Tomahawk and the nose cone assembly attain an altitude of approximately eighty kilometers, the entire system is despun, and the TP is ejected. From that point on, the TP and the rocket continue to separate at a rate of one or two meters per second until they reenter the lower atmosphere. During the coast to peak and the fall back to earth, the TP rotates freely with an end-over-end tumbling motion. In order to evaluate the data received from the mass spectrometer experiments, it is necessary to measure the attitude of the TP as it tumbles in the thermosphere. The measurement of the attitude is accomplished by the TP aspect system which is made up primarily of a solar or a lunar aspect sensor. By correlating these optical data with the mass spectrometer output, a complete attitude analysis can be performed. The purpose of the present report is to describe the motion of the TP, the aspect system, and the attitude analysis.

2. MODEL OF MOTION OF THE TP 2.1. ANALYSIS OF A TP EJECTION Before the TP can be ejected, the spin rate of the rocket must be substantially reduced. Two despin mechanisms are employed to reduce the spin rate from its nominal value of 6.6 revolutions per second. The first, called a "yo-yo despin module" (Fedor, 1961), is illustrated in Figure 1. There are two yo-yos, each of which consists of a 0.42-pound weight fastened to a cable 1/16 inch in diameter and 70 inches in length. The two cables are hooked onto a cylindrical housing at the base of the nose cone. They are wrapped in grooves 2.75 turns around the housing, opposite in direction to the anticipated spin of the rocket. Immediately before the ejection, the release of the weights causes the cables to unwind until they finally become unhooked. The increase in moment of inertia of the system causes the spin rate of the rocket to be reduced by a factor of about 8. The spin rate of the rocket is further reduced by the opening of the clam shells. When the clam shells are closed, the moment of inertia about the spin axis is approximately 1.43 slug ft2. The moment of inertia about the spin axis of the rocket with fully opened clam shells is approximately 17.5 slug ft2. Consequently, opening the clam shells increases the moment of inertia about the spin axis by a factor of nearly 12 and reduces the spin rate by the same factor. The combined effect of releasing the yo-yo and opening the nose cone is to reduce the spin rate of the rocket by a factor of about 100. The ejection of the Thermosphere Probe begins with the opening of the nose cone. When the clam shells reach an included angle of 150 degrees, a latch releases the plunger spring. The plunger spring has a spring constant of 100 pounds per inch (K) and is initially compressed 4 inches (Xmax). Upon being released, the plunger is allowed to travel a predetermined distance where a stop (Xmin) restricts any further expansion of the spring. This action causes the TP to move coaxially to the rocket with an initial separation velocity determined by the energy provided by the plunger spring. The TP is caused to tumble by a moment imposed by the "Neg'ator."* The negator consists of a spring motor and a cable which reels into the motor and maintains an approximately constant tension in the cable. The negator motor is mounted in the base of the nose cone, and the cable is hooked onto the side of the TP as shown in Figure 2. Consequently, a moment is applied to the TP which is approximately normal to the longitudinal axis. This moment is applied until the TP has rotated sufficiently for the negator cable to become unhooked. *"Negiator" is a trade name, hereafter used as a generic term for simplification. 2

NOSE CONE YO -YO DESPIN MODULE ROCKET s,-SPIN OFOCKE Figure 1. The yo-yo despin module.

NEGATOR CABLE CLAM SHEL 2n PLUNGER Figure 2. Structure of the nose cone and the ejection system.

The deceleration force of the negator has three effects. First, the separation velocity is reduced. Second, the TP is caused to tumble, that is, to begin an end-over-end rotational motion. Third, the initial spin rate of the TP can be significantly changed because the moment imposed by the negator does not remain normal to the longitudinal axis of the TP. The spin rate of the rocket influences the first two effects and primarily causes the third. Both the tension force of the negator and the length of travel of the plunger (Xmax-Xmin) can be modified in order to control the rate of rotation and separation velocity of the ejected TP. Certain assumptions are made in order to predict the motion of the TP during its ejection. The rocket and the nose cone are assumed to be rotating at a constant rate about the z axis of a nonrotating coordinate system called the base system, shown in Figure 3. In this system the z ax is coaxial with A A the rocket, and the x and y axes are defined to form a right-hand orthonormal system. The negator cable is assumed to be without mass and to extend from the hook on the TP to the source of the negator cable which rotates at a:constant A AA rate (%o) about the z axis in the x-y plane. The negator cable exerts a force on the rocket during ejection. It is assumed that the acceleration of the rocket (and of the base system) is translational and that the moment imposed by the negator on the rocket is negligible. Under these assumptions, the separation velocity between the TP and the rocket caused by the plunger can be calculated relative to the center of mass of the system. Let M1 be the mass of the TP, M2 be the mass of the plunger, and M3 be the mass of the burned rocket and nose cone. An instant before the plunger reaches the end of its travel, the TP and the plunger have a velocity V1 relative to the center of mass system, while the rocket has a velocity V2 as shown in Figure 4. 5

CENTER OF GRAVIT Y HOOK TP U NEGATOR CABLE z SOURCE OF THE NEGATOR CABLE BASE Figure 3. The negator causing the TP to tumble.

VELOCITIES JUST BEFORE PLUNGER REACHES X MIN -- V2 VI — _ I I I I L/ TOMAHAWK MS3 PLUNGER M2 TP Ml I I I I V3 V v I VELOCITIES JUST AFTER PLUNGER REACHES XMIN Figure 4. Velocity imparted by the plunger spring as viewed from the center of mass. By conservation of momentum (M1 + M2) V1 M3 V2, (2.1) and by conservation of energy 1 _ 1 K(X2 _ x2 ) = (M1 + M) V1 + - M3 V2 (2.2) 2 max min 2 2 An instant after the plunger reaches the end of its travel, the TP continues at V1, while the rocket and plunger have a velocity V3. By conservation of momentum M3 V2 - M2 V1 = (M2 + M3) V3. (2.3) Equations (2.1), (2.2), and (2.3) are solved simultaneously with the result

KM3(X2 - x2) max min.4 V1 = (Ml + M)(Ml + M2 + M)' (2.4) K(M1 +M2)(X2 - X2) max min V2 M3(M + M + M)' (2.) V = M3V2 - M2V1 (2.6) M2 + M3 The TP and rocket will separate at the rate V = V + V3. (2.7) A A A A coordinate system called the TP system with principal axes u, v, and w A is rigidly attached to the TP. The w axis is along the longitudinal axis of A A the TP. The u and v axes are defined to complete a right-hand orthonormal system, and the origin is at the center of gravity as shown in Figure 3. The TP is represented in this system as having a point mass (M1) at the origin and as having an inertia tensor with principal moments IT, I2, and 13 about the u, v, and w axes, respectively. The path that the negator follows from the rocket to the hook depends upon the position of the TP and of the source of the negator cable. If the line from the source of the negator cable to the hook does not pass through the TP, then the cable is assumed to follow a straight line. Otherwise the cable is assumed to follow the shortest path around the surface of the TP to the source of the negator cable. Define H to be the vector from the origin of the TP system to the hook and E to be the vector from the origin to the point where the cable leaves the surface of the TP. Note that E = H if and only if the cable is not wrapped around the surface of the TP. Let F be the vector which represents the force exerted by the negator cable on the TP. Then F is directed along the negator cable from the point where the cable contacts the surface of the TP, and it has a magnitude equal to the tension force of the negator. ~ A A A F = fix + fy + f3z (2.8) 8

Let T be the vector which represents the moment imposed by the negator on the TP. -* - - A A A T = E x F = TU + T2V + T3W. (2.9) Let R be a vector from the origin of the base system to the origin of the TP system. ~+ rA A A R = rlx + r2y + r3z (2.10) Then the translational motion of the TP and of the rocket is described as a function of time (t) by d2r f f n n n M M2 + for n = 12,3. (2.11) The rotation of the TP system is described in terms of a total rotation ~A A A vector (Q), and the axes of the TP system, u, v, and w. A A A W= + w W2V +WU3W. (2.12) U = U1X + U2Y + U3Z, A A A A V = V1X + V2Y + V3Z A A A A W 2 W1X + W2Y + W3Z (2.13) The transformation matrix from the base system to the TP system (TMX) is Ul U2 U3 TMX = vl v2 v3 (2.14) W1 W2 W3

The rotation of the TP system is described as a function of time by Euler's equations (Equations 11-7, Symon, 1961) dw __ M1 - (I3 - I2)(233 dt I1 M2 - (Il 7 i3)WlPQ3 dt 12 d = M3 - (I2 - I 11)w (2.1) (2.15) dt I3 A A A Finally, the time derivatives of u, v, and w are computed by using the equations du n dt Cl3n - nC2Wn dt dv n = -clwn - C3un dt dw n dt = e2un - elvn, (2.16) dt for n = 1, 2, and 3 (Equations 7-32, Symon, 1961). It should be noted that there are only three independent conditions described by these nine equations and that they are consequently redundant. The entire system of coupled differential equations is solved simultaneously by using Hamming's method of numerical analysis (Hamming, 1962). The initial boundary conditions for this analysis are rl (t=O) = r2(t=O) =0, r3(t=O) distance from base of plunger to the center of gravity of the TP, dr, dr2 = 0 dr(t ( d t) dr3(to) = o, 10

TIX (t=O) = l 1 02 O 0 L (t=O) = w (2.17) The solution of motion of the TP is presented as a computer tabulation of certain parameters at uniform intervals in time (see Figure 5). These parameters are as follows: A DISTANCE is the z coordinate of the center of gravity of the TP in feet. VELOCITY is the separation velocity of the TP and base coordinate systems in feet per second. TUMBLE RATE is the magnitude of the component of Q perpendicular to the longitudinal axis in degrees per second. SPIN RATE is the magnitude of the component of Q parallel to the longitudinal axis in degrees per second. AXIS THETA is the e-spherical coordinate of z in the base system in degrees. AXIS PHI is the 0-spherical coordinate of z in the base system in degrees. WRAP ANGLE is the 0-spherical coordinate of the source of the negator cable in the TP system in degrees. CABLE LENGTH is the length of negator cable from the hook on the TP to the source of the negator in the x-y plane in feet. RELEASE ANGLE is the angle between the cable and the w axis in degrees. When the release angle becomes approximately 90 degrees, the negator cable will be released, and the TP will continue rotating at constant spin and tumble rates. The computer solution is structured as though the negator cable remained attached to the hook. 11

EJECTION SIMULATION FOR NASA 18.52 AND NASA 18.53,, —NE-GATOR -FORCE?o000 POUNDS HOOK VECTOR 0.315 0.0 1.520 FEET....PLUN'GER TRAVEL 1.01n INCHES INERTIA TENSOR 3.120 3.120 0~090 SLUG FT FT INITIAL WRAP ANGLE 2n~orln OEGRFES TP MASS 2.2mn SLUGS -TIMF DISTANCF VFLOCTTY TUMRATE SPINRATE AX7S THFTA AXIS PHI WRAP ANGLE CABLE LENGTH RELEASE ANGLE n.? 2..........2432 5.647? 3M,000 q..00 20.000 3.935 1.661 -n. S,~ 2. - 71 5.5-3. n. 583 29.439, ).'INm15 35 2.72 19.956 4.234 1.550 ~.10n 2.9q).......... 5,~5523 1.........1,67......28,q13......n 8 MM340 4...___.5....._~512.......19.45_6'~. t 5~ 3.267 7~~~ ~~.464 1. 75 2 841 8 n 131 34.70 1, 1. 8. n.?nn 3.537 5.4n5 2.339 27.949 m.234 354.721 19.329 5,057 1.319 n. 25n 3.8o6 5. 346 2.93 n 27.501) n.365 355. 359 18.995 5. 325 1 ~284.rn 4. 972 15. 87 3.52 5 27.A67 %o526 355. 986 18. 633 5.590 1.277,N.35" 4.3'34' 5. 228 1.25 26.646.....n-718...356:6n1....18:2 6'9'......5':85.........3051.4011 4.594 5.168 4.733 26.231 0,939 357.205 17.929 6.110 1.375 n.45' 4.951 - 5.199 5.349 25,817 l.191l 357.798 17.636 6,365 1.488 o.R. 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(87 21.5110 11.487 16.433 1..7n'~ 1n. 41 3 3.63n 27. 794 -?,n5 1! q938 q,426 21,197 11,653 179651 ]. 75r- 1n.494 3.571?9.18 R -4.48 5 21. 358 q. 754 20.789 11 ~814 180934 i. 8 rr' 1). 671 3.512 3. 6 4 -7.r'l18 22.848 1 0. 071?O. 20.70 1..._,97.0 _...20.285_'.8 5' 1q.945 3.451 37.,153 -9,638 24,413 10.373 19,636 I2,1Z1 210706.gPP 1.'"1]6 3.'A94 33.728 -12.325 26. n55 Im. 665 18.883 12.268 23.202' 1.o",!1.194 3. 335. m S. 36 R - 15.58 27. 778 Im. 944 018.7 1 24824.775. —.nO'" 11.349 3. 276 -m 7. 97'Si - 17. 80)9 29.584 11.211 17.0.')7 12. 543 269430 2.05n 11~ 512 3.7~17't8.851T -20. 549 310477 11 0466 1 5.884 12.673 28.170 ~. InMn 11.671 3.158 4n. 6q9 -2 3.240) 330462 11.~709 14.637 { 2.279 30*0000 2.1 5'~ 71.827..............3.nqq.... 42.61 7 -25.843 35'540.... 11.939-........{. 3.270 12.914.4 6173125843.35950211493 2.2nn 11. 981 3.1'39 44.61fn -28.311I 37.717 12.156 11.790 13. 025 33.946 2.~25n 12.131 2..q81N 46.67 7 -30.596 39. 995 12. 362 10.~202 1 3.130 36.071 2.~3nn 12. 279 2.9C21 48. 82n -32. 643 42.378 12.~555 8:*519 13.227 38.304 2..3in 12.424 2.862 91.037 -34.399 44.871 12.737 6,751 13.318 40.6S0 2.4nn 12. 565 2.8Rr3 53. 329 -35.807 47.477 12.907 4.915 13.402 43.114 2.45n~ 1.n 2.744 55.694 -36.810) 50.200) 13.n6 3.130 13.479 4.0 -'-5"r 12.84n 2. 685 58.13n -37.356 53b043 13.215 1.115 13.549 4804{4k-.5 5', 12.972 2.626 60. 63 6 -37.397 56.010 13.353 -0.806 13.610 51.261'. 6on 13.7n2 2. 568 63. 20 7 -36.891 59.1 M4 13.482 -~2.707 13.665 540245 -2.65M 13.229 2.5n9 65.839 -35.808 62.328 13.602 -4.561. 13,711 570373 2.'70n 13. 353 2.415a1 68, 52 8 -340128 65.685 13.3715 -60340 13. 750 60.649 2.75~( 13.474 2.391 71. 266 - 31o 8 4 8 69.179 13.82) ~ 8.014 13.781 640077 2. 8nO) 13,592 2.332 74.047 -28,979 720810 130919 -9.555 13.804 670663?J. 8 56 13. 707 2. 273 76.86n - 25.553 76.5 81 14.013 -10.935 13. 819 71.412?.qnn 13.819 2. 214 79.696 -21.617 80.494 14.104 -12.129 13,.826 75.327 2.95n I.IJ. 29 2.155 82.543 - 17.23 9 84.549 14.193 - 13.112 13.827 790413 3~1Onn 14. n35 2. 096 85.38 6 -120 502 88.747 14.282 -13.~867 13.820 830674 3. 050) 14.~138 2. 037 88. 210 -7.50 8 93.086 14. 372 -14.376 13. 808 88.112 3, 1 nOi 14. 239 1. 978 90.996 -2. 369 97.565 14.466 -14.632 13. 789 920730 3..15n 14.336 1.~919 93.72 6 2 ~794 102.182 140566 -14.629 13. 767 97.529 3.20n 14.431 1.~860 96.376 7.85 6 106.934 149675 -14.370 13. 741 102,,509 3.25n 14.522 1.80t 98.92 5 12.692 111.816 14.797 -13.861 13.713 107.668. 3.3nn 14.611 *742 101*346 17.189 116.821 14.936 -13.116 13.685 113.002 3.3 50 14,696 1.683 1 03.61 5 21.243 121.944 15.097 -12.153 13.659 118.504 3.400 14. 779 1. 624 105. 704 24.776 127176 15.290 -10.992 13.637 124.167 3.450) 14.859 1 ~565 l07. 588 27.731 132.506 15. 525 -9.660 13.620 129.977 3.500) 14.936 1.50'6 109.242 30.083' 137.924 15.1&F~ -8.180 13.611 135.921 Figur'e.Computer' format for the mahematical solution of a ty'pieal ej ection of a TP.

2.2. DESIGN OF THE EJECTION SYSTEM The force of the negator, the length of travel of the plunger, and the initial wrap angle can be adjusted to control the rotation rate of the ejected TP. The initial wrap angle is the angle through which the TP is rotated inside the nose cone from the position where the hook and negator source are aligned. In general, it is desirable to minimize the spin rate and to choose a tumble rate suitable for the particular experiments in a TP. The ejection analysis and computer program can be used to adjust the ejection system and to achieve the desired tumble and spin rates. The first step in the procedure for obtaining the design parameters of the ejection system is to run the program with a series of data sets and solve the ejection with a nominal length of travel of the plunger and a variety of negator forces. To the first order of approximation the tumble rate at release is independent of the length of travel of the plunger. Consequently, the final tumble rate can be plotted versus negator force as shown in Figure 6. The negator force required for a desired tumble rate is easily read from this graph. Once the negator force has been specified, the length of travel of the plunger is chosen. The program is again run, and the ejection is solved for the chosen negator force and a variety of lengths of travel of the plunger. Plots of the extension of the negator cable and of separation velocity versus length of travel of the plunger can be made as shown in Figures 7 and 8. The extension of the negator cable is defined as the difference between the cable length before ejection and the maximum cable length required for an ejection. From these graphs the length of travel of the plunger can be selected so that the separation velocity is acceptable and so that the physical limitations of the negator are not exceeded. The third step in the design process is to solve for an initial wrap angle which will produce the desired final spin rate. The program is run again for a variety of initial wrap angles, with the negator force and length of travel of the plunger chosen in the previous steps. The final spin rate can be plotted as a function of initial wrap angle as shown in Figure 9. It should be noted that the final spin rate is also dependent upon the initial spin rate, so that a family of such curves can be constructed, each for a different initial spin rate. By the assumption of a nominal value for the initial spin rate, an initial wrap angle can be chosen from this graph. 13

200- EJECTION ANALYSIS NASA 18.52 180 NASA 18.53'160 w < 140 z 120 100 J 80 z 0 2 4 6 8 NEGATOR FORCE - LB Figure 6. Final tumble rate versus negator force for a typical TP. u 14 IJIw ~O~ 12 / EJECTION ANALYSIS Iz< / NASA 18.52 C 10 NASA 18.53 Co / NEGATOR FORCE - 2LB,< 8 - z 0 I 2 3 TRAVEL OF PLUNGER-IN. Figure 7. Extension of the negator cable versus length of travel of the plunger for a typical TP. 14

LC) (/3 4 LL), 3 L / EJECTION, ANALYSIS / / NASA 18.52 > / NASA 18.53 NEGATOR FORCE - 2LB ~ 2 <II 0 I 2 3 TRAVEL OF PLUNGER - IN. Figure 8. Separation velocity versus length of travel of the plunger for a typical TP. C) X 20 20 DEG/SEC ~0 ROCKET SPIN RATE 4 0 w EJECTION ANALYSIS -J Ic -lo _ NASA 18.52 cr/ ~NASA 18.53 -20 - NEGATOR FORCE - 2LB I- TRAVEL OF PLUNGER-I IN.: -30 30 DEG / SEC z ROCKET SPIN RATE a. -40 n -30 -20 -10 0 10 20 30 INITIAL WRAP ANGLE- DEG Figure 9. Final spin rate versus initial wrap angle for a typical TP. 15

2.3. MOTION OF THE TP AFTER EJECTION After separation from the nose cone assembly, the TP is considered to be free from all external forces except for the earth's gravitational attraction. Under this assumption, a mass point trajectory can be calculated and fitted to data obtained from radar measurements of position versus time. The result is a flight history of position and velocity as a function of time. The rotational motion is calculated under the assumption that the TP rotates freely about its center of gravity. This motion is characterized by the spin rate ((3), the tumble rate (t = ct + 2+- ), the moments of inertia (I1, I2, I3), and the angular momentum vector (i) of the TP. In general, it is assumed that Ii = I2 > 13, and that after ejection, t remains constant throughout a flight. As the TP rotates freely, its longitudinal axis generates a conical surface symmetrical about ~, as shown in Figure 10. This cone of motion is described by the angular parameter (b), called the coning angle. The parameter 6 is defined as the angle between the cone of motion and a plane perpendicular to L called the tumble-plane, and is determined by I1t cot 8 = I (2.18) I3w3 L CONE OF MOTION VELOCITY VECTOR Figure 10. Motion of the TP in free space. 16

3. ATTITUDE SENSING INSTRUMENTS In order to measure the attitude of a body in space, it is necessary to have information concerning the orientation of a body relative to two linearly independent vectors (directions). In the case of the TP, aspect measurements are normally made with reference to an optical vector toward the sun (S), or the moon (M), and to a second vector (V) of the translational velocity of the TP. Therefore, the TP has two independent aspect sensing instruments: the first is an optical sensor and the second is a pressure gauge. These aspect sensing instruments measure two angular parameters which are used to solve for L. The optical aspect sensor measures the first parameter, the angle between L and the optical vector S or M, called e or 0, respectively. The second parameter is an angular phase difference bLeween tLhe outputs of the two sensors and is called y. 3.1. SOLAR ASPECT The solar sensor is the primary attitude sensing instrument when a Thermosphere Probe is launched during the day. It is an optical sensor which measures the orientation of the TP relative to the solar vector (s) and consists of an array of light-sensitive elements mounted in the center section of the TP. The field of view of the solar sensor is a. 360 degree fan which is symmetrical about and approximately normal to the longitudinal axis of the TP (Figure 11). To be more specific, the field of view subtends the solid angular region which is bounded by two conical surfaces symmetrical about the longitudinal axis. In terms of the TP coordinate system defined in Section 2, A A the field of view contains a plane parallel to the u-v plane and extends about one degree above and below it. Therefore, the field of view is approximated A A by a plane parallel to the u-v plane which is called the solar sensor plane. As the TP tumbles, the sun will enter and leave the field of view twice during each tumble. When the sun enters the field of view, the solar sensor will trigger and read out a coded digital signal. This digital signal contains information about the value of the ~-spherical coordinate of S in the TP system. Consequently, at the time of each solar sensor readout, - (expressed in the TP system) is in the u-v plane, and its ~-position is known. 17

A~ U S~~~tAR ~ PLANE \ ANG I - 30LARSNSO - ~~FIELD OF VIEW Figure 11 Measuremn a t e olarsensor.

3.2. LUNAR ASPECT For night launches, a lunar sensor is used in place of the solar sensor. The lunar sensor is constructed by mounting the light-sensitive base-emitter junction of a silicon transistor behind a glass plate which has been masked by a series of concentric opaque rings. The lunar sensor is mounted in an end of the Thermosphere Probe with the glass plate perpendicular to the longitudinal axis. Consequently, the lunar sensor has a field of view which is a series of concentric cones symmetrical about the longitudinal axis (Taeusch and Grim, 1967). As the TP tumbles, the moon passes through the field of view, and the sensitive area is alternately shaded and exposed to the moonlight. The resultant lunar sensor output indicates the minimum angle between the TP axis and the vector toward the moon, and also indicates the time during a tumble at which the minimum angle occurs. The minimum angle is interpreted as T/2 + ELM + 6, and the angle y can be calculated from the time relationship of pressure maxima and the times of minimum angle to the moon vector. The net result is that the mathematical analysis is nearly identical to the case of solar aspect. 5.3. VELOCITY VECTOR ASPECT The second aspect sensing element in the Thermosphere Probe is an axially mounted pressure gauge. An omegatron mass spectrometer (Niemann and Kennedy, 1966) is mounted in an end of the TP with its orifice symmetrical about the longitudinal axis. Although the primary function of the omegatron is to measure atmospheric density, its output also reflects the angle between the velocity vector (V) of the TP and the longitudinal axis. The output of the omegatron is at a local maximum approximately when the angle between V and the longitudinal axis is at a minimum. This relation allows for the calculation of I, the tumble angle from the point of minimum angle of attack to the point of the nearest solar sensor output. 3.4. EARTH NORMAL ASPECT It is also possible to perform the attitude analysis with reference to the optical vector and a vector toward the center of the earth called the earth normal vector. In this case a horizon-sensing instrument is mounted in an end of the TP. When the longitudinal axis of the TP crosses the horizon, the thermal radiation of the earth's lower atmosphere is detected, and the sensor "turns on." When the second horizon is crossed, the sensor "turns off." The longitudinal axis of the TP is assumed to be at its minimum angle to the earth normal vector, midway between the times of turn on and turn off. Consequently, the earth normal vector can be used as a reference vector for determining attitude in the same manner as the velocity vector, and the mathematical analysis is identical. 19

4. PARAMETERS OF THE ATTITUDE SOLUTION 4.1. TUMBLE-PLANE COORDINATES Solving for the attitude of the Thermosphere Probe necessitates solving for the total angular momentum vector L. In order to arrive at a mathematical solution, certain parameters are measured from the outputs of the optical sensor and from the pressure gauge. These parameters are easily visualized if the rotating TP is viewed in the special coordinate reference frame called the A tumble-plane coordinate system (Figure 12). Let the vector be the local z axis, and the x axis be defined as the projection of i (or A) into the tumbleplane. The y axis is constructed by requiring a right-hand orthonormal coordinate system: A z y Lx S A A A x - yxz. (4.1) Then in this system A A = x sin E3LS + cosLS (4.2) LS LS A A A V = x os sin eLV Y siin "V sin eLV + cos e (4 3) V LV V LV LV where ELS is the angle between t and A, A A eLV is the angle between L and VY A XV is the ~-spherical coordinate of V. In this coordinate system, the rotational motion of the longitudinal axis of A A A w = x cos at(t - to) cos 6 + y sin Wt(t - to) cos 6 + z sin 6 (4.4) 20

where wt is the tumble rate. Then define the tumble angle as WLO = t(t - t0) which is the c-spherical coordinate of the TP in the tumble-plane system. A A L=z TUMBLE- PLAN E A y x /? LS "LV Fiure 12. Tumble-plane coordinate system. This mathematical representation can be applied to the models for the field of view of the solar sensor and for the conical motion of the TP in order to predict the relative time that the solar sensor will trigger. If the motion of the TP were planar, then the solar sensor would trigger when = + /2. In general, the motion is conical with a coning angle 6, and the solar sensor field of view is not necessarily planar. It will be assumed that the solar sensor triggers when the angle between w and S becomes (./2) - f, where f is the angular deviation of the boundary of the solar sensor field of view L Y from the solar sensor plane. Nonzero values of 6 or of f cause the solar sensor to trigger at a tumble-plane angular deviation (t) from the sismle case of C can be calculated in the following manner: at the time of a sun pulse, 21

A A w.S = sin f, A A A A w = - x sin E cos y + y cos C COS e + z sin 5, sin f = - sin E cos 3 sin 8LS + sin b cos eLS Consequently, sin = cot 9LS tan 6 - sin f/cos 5 sin.LS (4.5) The parameter e is a signed angle and - c/2 < e < + t/2. Normally e is small enough so that it can be neglected in the aspect computations. A A L=z A h ~~A W x Figure 15. Solar sensor StNSE-2.2. THE PARAETERUN PULSE The primary parameters in the solution for attitude can be expressed in i ~ ""~" \!~~~USL-LN Fiur 1. olr enortrggr/g 22

The parameter y is defined as the angle between the projection of V into the tumble-plane and the vector w at the time when LO = + t/2, with the restriction that - /2 < y < t/2. In other words, 7 = J/2 -iV or Y = 3/2 - V'(4.6) whichever is smaller in magnitude. The angle y can be measured directly from the outputs of the pressure gauge and the solar sensor. The axially mounted omegatron sees a local maximum pressure once during each tumble. This occurs approximately when the angle between w and V, called the angle of attack (c), is minimal, that is, when Because of the relation between cO and the output of the omegatron, te time when PLO 4V can be measured for each tumble. Then y is calculated by t2 ime of nearest sun pulse - time of omegatron maximum] tumble period (4.7) Certain small corrections must be applied to this direct estimate of y. The omegatron response is not instantaneous since the ambient density changes rapidly as the TP moves along its trajectory, and E may not be negligible (see Appendix A). For the case of lunar aspect, 7 can be computed in a similar manner. The angle 4V is calculated from [lime of center of moon pulse output - time of omegatron peak] V tumble period (4.8) and then 7 is given by Equation (4.6). 4.3. THE PARAMETER eLS The second important parameter in the determination of L is Se the angle between L and S. Its value is determined directly from the output of the solar sensor. In order to derive the expression for 9SL first assume that the TP is not rolling, that is, c 3 = O. In this case a is zero, and consequently e is zero. The angular momentum vector is continually in the plane of the sun sensor. 23

A Because of the zero spin condition, L will be located at a constant angle (%o) on the sun sensor angular scale. At time tn the sun moves into the plane of the sensor, and the sensor reads out its angular location In. Half a tumble later at time tn+l, the sensor reads out an angle In+l. Then n o LS n+l 0 LS' + 2e = -. (4.9) LS n n+l If the TP is rolling slowly, Equation (4.9) can be modified to account for roll: ~ LS n 4 En + (l n+l tn) (4.10) In general, c3 is sufficiently small so that 5 and E can be neglected, and the expression for e will yield valid results. If these simplified conditions do not exist, then a more detailed analysis is necessary to account for large E and large 6 (see Appendix B). It should be noted that 0 cannot be distinguished from ET - 8 unless additional information is available. If there were some way that a sun pulse occurring at TLO - n/2 could be distinguished from one at L + /2, then the exact 0LS could be determined. If a sense-l sun pulse is Refined as one which occurs when - nit/2, and a sense-2 sun pulse as one which occurs when LO + /2, and a sense-l sun pulse occurs at time t, then n o LS - - (t - t n+l o LS tn+l n and 20 = " - -(tn -t). (4.11) LS n n+l r n+l n Therefore, if the sense of the sun pulses is known, the exact 0LS can be determined. Otherwise, 0LS is known to be either an angle or its supplement. 24

SOLUTION FOR ATTITUDE 5.1. SOLUTION FOR LS A Once the angles y and eLS have been determined, the solution for L is performed in another coordinate reference frame called the sun-velocity coordinate system, as shown in Figure 14. It is defined as follows: A A Z = S Y - SxV, A A A X Y x Z. (5.1) Make the following representations: A A A V = X sin S + Z cos ae (5.2) A A A A L X sin eLS cos co LS +Y sin eLS sin 0LS + Z cos eLS. (5.3) Assume that the TP is tumbling is a plane; then a sun pulse will occur when w(sun pulse) = + S x L - +(X sin mLS - Y cos LS). (5.4) Define an intermediate vector A: A A A = V x L. (5.5) It can be seen in Figure 15 that both w(sun pulse) and A are in the tumbleplane, and that ILA I sin -y = *.w(sun pulse)l; or [w(sun pulse).A] = A.A sin27.(56) 25~~~~~~~~~56

A A S z A V ^ A Y I x~~~~~~~~~~~~~~~ Figure 14. Sun-velocity coordinate system. A /=z ^ TUMBLE-PLANE A = = I I y~~~~~~~~ A I Y Figure 15. Aspect solution in the tumble-plane coordinate system. 26

A A A A Vx L A X(-cos eSV sin eLS sin )LS) A + Y(cos GSV sin eLS cos LS cos LS sin SV A + Z(sin EsV sin LS sin LS (57) w(sun pulse).A - cos LS cos eLS sin eSV - cos eSV sin eLS. (5.8) By substituting Equation (5.8) into Equation (5.6) cos2~LS co LS sin2SV + sin2LS cos20SV - 2 cos 4LS sin SV cos SV s in LS co LS equals sin24LS sin2eLS sin2y + cos2LS cos 2sv sin2 GLS sin 2y + sin2eSV cos2a LS sin2y - 2 cos ~LS sin SV s SV sin LS cos 0 sin. (5.9) Then by substititing sin2 = 1 - cos2 Equation (5.9) becomes a quadratic in cos LS. The solution of His quadratic is -b ~+ 2 - 4ac cos - (5.o) LS 2a where a - sin2Sv (sin2y sin2eLS + cos2L ) b = - 2 cos 8SV sin eSV cos OLS sin cLS 2 27

c = cos in -sin 2S- sin2 sin - C sin e Sv LS LS LS BV The above expression yields two distinct values for cos 1 which in turn yield LS four solutions for 4LS' In order to resolve the ambiguity of four solutions, two or more separate aspect determinations are performed for a flight. By comparing the sets of solutions for DLS' extraneous solutions can usually be eliminated and the accuracy of the correct value of f can be improved. Once LS is known, then L is given by Equation (5.3). L A second method for finding the value of ~ involves exploring the functional relationship between y,, TLSP and LS' Te previous method approached this problem by considering L to be a multiple-valued function of y and LS* Perhaps a more natural approac would be to consider y a single-valued function of LS and e LS LS LS Y(LSBeLS) = y(L). (5.11) This relationship holds over the range - LS - and o < *s <- 2=. -LS - It is also noteworthy that this function has the following symmetry: 7(L) -). (5.12) If eLS were held constant and equal to the measured value, then y could be considered a function of L only. The second method involves fixing the value of eLS and computing the value of y(l$ ) for values of a taken at one degree increments over the entire range of 1LS. Then LS can e looked up directly from the resultant tabulation of y( S) which is plotted in Figure 16. The actual computation of y is done on a digital computer by usingAthe following method. Given values of ~ and 9 the cartesian components of L are computed. Then a matrix translormation for a tumble-plane coordinate system is constructed, and V is transformed into this coordinate system by a matrix multiplication. The value of XV is computed and then y is given by Equation 28

TYPICAL PLOT OF Y VERSUS 0LS SOLUTION FOR 0LS WITH VALUE OF r MEASURED EQUAL TO 40 DEG ~80 ~60 ~40 20 0 40 — ~60 80 0 0 0 - I I I I I I I I I I I I I0 40 80 120 160 200 240 280 320 360 0 LS DEG Figure 16. Typical plot of y versus 4LS

A The mathematical methods of solving for L can be applied when the earth normal technique of evaluating the attitude of the TP is employed. This technique requires a horizon sensor in the TP, and the analysis is identical to the previous analysis with the earth normal vector (N) substituted for V. Both the velocity vector method and the earth normal method may be employed to evaluate the attitude of the TP. A The velocity vector method has the advantage that V changes significantly during a flight. As V changes, the parameter 7 changes in a continuous manner. As a result, three of the four mathematical solutions for L will wander, and the correct solution will remain constant. Consequently, the redundancy in the solution for PLS is easily resolved. The disadvantage of the velocity vector method is that the measurement of y can have significant systematic errors. This problem is caused by the difficulty in measuring the time when a is minimal (see Appendix A). An ambient particle wind can also complicate this problem. The earth normal method has the advantage that horizon sensing instruments can resolve the earth's horizons to an accuracy of better than one degree. This resolution exceeds the accuracy with which 7 can be determined in A the velocity vector method. Consequently, the earth normal solution for L is the more accurate method. However, N does not change significantly during the time of flight, so that the redundancy of the solution for LS cannot be resolved by this method. The earth normal method also has the disadvantage that the horizon sensor, occupying an end of the TP, limits the space available for other instrumentation. Another disadvantage is that in sounding rocket applications the cost of horizon sensors can be prohibitive. 5.2. CALCULATION OF ANGLE OF ATTACK A After 6 and L have been determined, the angle of attack (ca) from the TP longitudinal axis to the velocity vector (V) is calculated. The usual technique for reducing TP data requires only the minimum angle of attack during a tumble (C~in), which is given by the relation sin (amin +) = LV (5.13) min The angle (aRF) from any vector fixed on the body of the TP (F) to any other reference vector (R) can be calculated after the cone of motion is known. The general method for calculating ORF is to compute the attitude of the TP and then to express the TP fixed vector and the reference vector in the same coordinate system. The first step is to construct the axis vector of the TP,j) as a function of time. If a sense-l sun pulse occurs at time tn then w is given as a function of time (t) by j0

A A A A w(t) x sin wt(t - tn) cos y - y cos ct(t - t ) cos ~ + z sin. (5.14) where x, y, and z form theAtumble-plane coordinate system. Now, define a coordinate system with axes X, Y, Z. where A A Z = w Y = wx L A A A X Y x Z. ( ) The vector fixed with respect to the TP (F) is represented by A A A A F X sin O cos G + Y sin O sin F + Z cos GF (.16) F F A A where OF is the angle between F and w, and where O is the O-spherical coordinate A of F. The value of eF is a known constant, and O is calculated as follows: let OF be the angle on the sensor scale of the projection of F into the solar sensor plane. Then F -n -XS +L 3(t -t), (517) A A A A where eXS = cos-l(X.S). Therefore, if R is the known reference vector, and F has been calculated, R =- cos-(R.F ) (5.18) 31

APPENDhIX A MEASUREMENT OF y The parameter y is defined as the tumble angle from a minimum angle of attack to the nearest ideal sun pulse, and ranges from - t/2 to + ir/20 Its measurement from telemetered aspect and pressure gauge data is complicated by the following: 1. Sun pulses are displaced by the angle C from their ideal location. This error can be corrected by using Equation (4.5) to calculate c. 2. The finite acoustic conductance of the antechamber of the omegatron causes a time delay in the pressure gauge output which is strongly dependent upon the dimensions and the shape of the antechamber. 3. The change of ambient density with altitude shifts the peak pressure in time away from the point of minimum a. The amount of shift is dependent upon the velocity of the TP and upon the rate of change of density with altitude. This time shift can be calculated on the basis of a closed source, the pressure gauge chamber (Ainsworth, et al., 1961; Schultz, et al., 1948). ni - na Ta/Ti F(S). (A.1) 2kT S = S cos a = V cos a/ a-ka. (A.2) F(S) = exp - S2 +T S[l + erf S]. (A.3) The parameter S can be represented as a function of time where to is the time of a peak pressure and ct(t - to) is much smaller than T/2. 2kT S = V cos mino s s ct(t - t)/ m (A.4) min t m In this representation, at t = to, the TP is at the minimum angle of attack. If the pressure gauge shows a maximum pressure at time t = to + At, then At will be the time shift of an omegatron peak. To solve for At, set dn __ (At) o 0 dt 32

By differentiating Equation (A.4) d~nT dS S S + wt tan wt(t - t ) d+ a (A dt = o 2-' dF(S) = zT (1 + erf S) (A.6) dt dtS FS) (1 + erf S)9 + tan (t - t ) + I dt F(S) V a 2dtj (A.7) dn din n dinT 1 i a a =O + ni dt dt 2dt S,~~~~ d i n T F(S- (1 + erf S) + at tan t(t + t (A.8) Let us assume that At at << 1 din n dinT F(S cos i a + a dinT o min dt 2dt g a + - tn =(t At. (A. 9) F; S cosc () 1 + erf Scos 2dt t o min o min By the ideal gas law and the hydrostatic equation dinT din n a a = + (A. 10) dh dh kT a din n dinT V dT a a a ( dt 2dt T k 2dh Therefore, dT V F(S cos C )mg + a V dT At - z 0 min k 2dhj z a w2T S cos [ + erf S cos ] t t tao min o min (A 12)

Where h = altitude g = gravitational acceleration Vz = TP vertical velocity V = TP total velocity m = molecular weight of gas Ta = ambient temperature =t tumble rate in radians per second So = speed ratio = V/(most probable thermal particle velocity) S = V(normal to orifice)/(most probable particle velocity). 34

APPENDIX B GENERAL eLS ANALYSIS LS Let a sense-1 sun pulse occur at time tn, sense-2 at tn+l, and sense-1 again at tn+2. Represent the TP vector as A A A W(t) - = w (B.1) n n n+2 A As in Figure 17, define a vector C by A A A C =L x w A A A A C(t) = Lxw = C. (B.2) n n n AA L=z MOTION OF TP MV k t ~~~~~~~~~~~f/ A \ - Wn K E 2 TUMBLE-PLANE Figure 17. General 0LS analysis. 35

Consequently, C w = O. (B.3) n n Sew = o. (B.4) n A Define the angular parameter 8* as the angle between S and C. Then Cos 9* 3 S*C. (B.5) where, in the tumble-plane system A A A S = x sin LS + z cos LS. (B.6) A A A C x cos c - y sin E. (B.7) n cos O* = sinLS COS c sin E = C (B.8) LS cos e Therefore, if both O* and c can be measured, LS can be calculated from Equation (B. 8). Let us assume that eLS < A/2. Also assume that the TP is spinning with a constant spin rate cW3, and that the solar sensor reads out the angle In at tn, ~n+l at tn+l, and ~n+2 at tn+2. Now since Cn is situated in the plane of the solar sensor, it will have an angular position in the sensor scale. Call this position fc' Consequently, C n n c n+l n+l 36

c n+2 n+2 c = + cL3(t - t), cn+1 cn n+1 n n+l n =r + +3 (t -t )(B. 9 c c n+2 n+l (B.9) n+2 n+l Now solve this system for e*; 4e* = n3(tn -2tn+ + t ) - (n- + ) (B.10) n 2tn~ n+2 n n+2 Note that this analysis is valid only if OLS < n(/2. Also note that all arithmetic with angles is done with a modulus of 23r. In the case where /2 < LS e < 4 - = -E* c n n c n+ l - ~ = -e* (B.ll) c n+2 (B.) n+2 The resulting solution is -4e* =w3(t'n 2tn+l tn+2 n n+l n+2 (B. 12) 37

REFERENCES Ainsworth, J. E., Fox, D. F., and LaGow, H. E., Measurement of Upper-AtmosPhere Structure by Means of the Pitot-Static Tube, NASA Technical Note D-670, Goddard Space Flight Center, February, 1961. Fedor, J. V., Theory and Design Curves for a Yo-yo De-spin Mechanism for Satellites, NASA Technical Note NASA TN D-708, Goddard Space Flight Center, August, 1961. Hamming, Richard W., Numerical Methods for Scientists and Engineers, McGrawHill Book Company, New York, 1962. Niemann, H. B., and Kennedy, B. C., "An Omegatron Mass Spectrometer for Partial Pressure Measurements in the Upper Atmosphere," Review of Scientific Instruments, V. 37, No. 6, p. 722, 1966. Schultz, F. V., Spencer, N. W., and Reifman, A., Upper Air Research Program, Engineering Research Institute, University of Michigan, under contract with Air Materiel Command, Cambridge Field Station, Contract No. W-33038 ac-14050, July 1, 1948. Spencer, N. W., Brace, L. H., Carignan, G. R., Taeusch, D. R., and Niemann, H. B., "Electron and Molecular Nitrogen Temperature and Density in the Thermosphere," Journal of Geophysical Research, June 1, 1965, V. 70, No. 11, pp. 2665-2698. Symon, Keith R., Mechanics, Addison-Wesley Publishing Company, Reading, Massachusetts, Second edition, May, 1961, pp. 269-293; 450-451. Taeusch, D. R., Carignan, G. R., Niemann, H. B., and Nagy, A. F., The Thermosphere Probe Experiment, University of Michigan, Rocket Report 07065-1-S, March, 1965. Taeusch, D. R, and Grim, G. K., "Sounding Rocket Optical Aspect Determining System," NASA Reportable Items Disclosure, Contract No. NAS 5-9113, April, 1967. 38

UNIVERSITY OF MICHIGAN II 111 B3 II I 7 I9 E 1 15 l0 8 7922 11 1 3 9015 02082 7922