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NASA TECHNICAL NOTE IASA TN D-3117

MONTE CARLO APPROACH TO TOUCHDOWN DYNAMICS FOR SOFT LUNAR LANDING

by Robert E. Lavender George C. Marshall Space Flight Center HuntsvMe, Ala.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION X

.L38 IMS

WASHINGTON, D. C. - NOVEMBER 1965

LINDA HALL LIBRARY

3 3690 00585 0721

NASA TN D-3117

MONTE CARLO APPROACH TO TOUCHDOWN

DYNAMICS FOR SOFT LUNAR LANDING

By Robert E. Lavender

George C. Marshall Space Flight Center Huntsville, Ala.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $3.00

VTt If 5 i

TABLE OF CONTENTS

Page

SUMMARY i

INTRODUCTION i

VEHICLE CHARACTERISTICS i

INITIAL CONDITIONS 4

TOUCHDOWN DYNAMICS PROGRAM 9

RESULTS 11

APPENDIX A. TOUCHDOWN CONDITIONS . . . . 37

APPENDIX B. RANDOM NUMBER GENERATOR CONFIDENCE TEST . . . . 47

LIST OF ILLUSTRATIONS

Figure Title Page

1. Vehicle Configurations 17

2. Spring Constants for Vehicles I, II, and III . . . . . j. 18

3. Force-Stroke Relationship 19

4. Chi-square Frequency Function 20

5. Chi-square Distribution Function 21

6. Normal Distribution Function 22

7. Surface and Body Axes Orientation 23

8. Symmetric Heading Orientations 24

9. Stability Boundary for Vehicle 3 25

10. Stability Boundary for Vehicle 7 . . . 26

11. Lower Limit Probability of Stable Landing 27

i i i

Linda Hall Library ' Kansas City, MO.

LIST OF TABLES

Table Title Page

i . Geometric Proper t ies of LFV, LEM, and LLS 28

2. Geometric Proper t ies of Vehicles I, II, and HI 28

3. Limiting Forces for Type I Vehicles 29

4. Dynamic Scaling Factors 29

5. Initial Conditions from Chi-Square Distribution 30

6. Tabulation of Distribution Functions 31

7. Summary of Touchdown Dynamics Runs for Vehicle 1 32

8. Summary of Touchdown Dynamics Runs for Vehicle 5 34

9. Summary of Cases which Tumble 35

10. Sample Probability of Stable Landing 36

11. Lower Limit Probability of Stable Landing 36

IV

DEFINITION OF SYMBOLS

Symbol Definition

C Confidence coefficient

CSA Velocity c ross -s lope angle (angle between total horizontal velocity vector and principal slope direction)

C Crushing force in main s t ru t mp

C Crushing force in support s t rut sp

D Landing gear diameter

g Lunar gravity

k Vertical spring constant per leg

k Radius of gyration in pitch

L Height of the center of gravity

m Vehicle mass

n Number of cases which tumble

n Number of cases in sample

N Number of legs

p Sample probability of stable landing

p Lower confidence limit on probability of stable landing for binomial population

r Radius from vehicle 's centerline to main s t ru t attachment point

V, , Nondimensional total horizontal velocity ht J

V Nondimensional vert ical velocity v

V, , Total horizontal velocity ht J

DEFINITION OF SYMBOLS (Concluded)

Symbol Definition

V, Component of horizontal velocity in the direction of principal slope

V Vertical velocity

V Component of horizontal velocity perpendicular to direction of principal slope

Y Distance from bottom of vehicle to main s t rut attachment point a

Y Distance from landing gear foot pads to bottom of vehicle P

a , /3 , y Dynamic scaling factors shown in Table 4

y Lunar Slope

Vehicle's pitch angle referenced to the horizontal

Initial pitch angle with respect to lunar surface

P

0 Vehicle 's initial heading orientation with respect to principal slope direction

03 Equivalent heading orientation for three-legged vehicle

cp 04 Equivalent heading orientation for four-legged vehicle

MONTE CARLO APPROACH TO TOUCHDOWN DYNAMICS FOR SOFT LUNAR LANDING

SUMMARY

Results of analytical touchdown dynamics investigations a re presented which were conducted to obtain es t imates of the probability of stable landing for configurations with various landing gear d iamete r s . Both three-legged and four-legged vehicles a r e considered in the analysis . Dynamic scaling considerations a re taken into account so that the resu l t s a re applicable to a wide range of vehicle size and m a s s . A Monte Carlo approach is taken in the determination . of initial landing conditions.

Results indicate that for a given probability of stable landing, a t h ree -legged vehicle requires a landing gear diameter only slightly l a rge r than the diameter required for a four-legged vehicle. Therefore the three- legged vehicle 's landing gear should weigh l e s s .

INTRODUCTION

In the overall design and performance of spacecraft intended to soft-land on the moon, analysis of touchdown dynamics motion is an important pa r t . Such analysis in the past has been res t r ic ted to specific missions and vehicles designed for those missions such as Surveyor (ref. i ) , Lunar Excursion Module (ref. 2), Lunar Flying Vehicle (ref. 3), and Lunar Logistics System (ref. 4 ) . In general, the analysis for each vehicle has not been a complete paramet r ic study over a wide range of initial conditions, but ra ther a limited study with emphasis on what the vehicle can do under the worst set of conditions. The paramet r ic approach is not feasible because of the large number of pa rame te r s involved. For example, if a pa ramet r ic study of a par t icular vehicle is considered with only three variat ions of ten initial touchdown conditions, a total of three to the tenth power, o r 59, 049 combinations, is produced.

Presented in this repor t a re resu l t s of a general touchdown dynamics study in which a Monte Carlo approach is taken to the determination of initial conditions. This approach is more real is t ic than simply choosing the wors t -case conditions and does not require the prohibitive amount of analysis which a

paramet r ic approach would requi re . The vehicle configurations used a re determined in a standardized form, and the resul ts a re equally applicable to the Lunar Flying Vehicle, Lunar Excursion Module, or Lunar Logistics System, even though these vehicles vary greatly in size and mass .

Initial conditions have been determined using a Monte Carlo approach for 400 cases . For each case, the initial conditions were determined for 10 p a r a m e -t e r s : vert ical velocity, horizontal velocity, velocity c ross - s lope angle, vehicle pitch, bank, and heading angles, lunar slope, and the vehicle 's pitch, roll, and yaw ra tes at touchdown. The coefficient of friction between the landing pads and lunar surface is assumed to be infinite (no s l iding) . Eight generalized vehicles a r e analyzed for these 400 cases to determine whether or not the vehicle being considered lands safely or tumbles. The eight vehicles considered correspond to three-legged and four-legged vehicles with four variat ions in the ratio of landing gear diameter to center-of-gravity height. F r o m these resul ts , an est imate is made of the probability of stable landing on the lunar maria for each of the eight vehicles .

The author is indebted to Mr. John D. Capps, Computation Laboratory, who programmed the equations for the three-dimensional touchdown dynamics digital p rogram as well as the program used in the determination of initial conditions.

VEHICLE CHARACTERISTICS

The Lunar Flying Vehicle (LFV) is a small vehicle which has received consideration for lunar surface exploration after the initial Apollo landings. P r imary emphasis has been placed upon a vehicle having an 80-kilometer maximum range when carrying a payload of two men with their portable life support sys tems . The p r imary mission for such a vehicle was considered to be rescue back to the Lunar Excursion Module (LEM) from a disabled surface roving vehicle. A secondary mission was that of supplementing the roving vehicle by permitt ing flights into a reas which a re inaccessible to a roving vehicle.

The Lunar Logistics System (LLS) is a large cargo-car ry ing vehicle which has received study as a means of delivering 15-ton payloads to the moon in support of a lunar base .

Even though the three spacecraft (LFV, LEM, LLS) vary greatly in size and mass , all three vehicles can be reduced to approximately the same nondimen-sional geometry as shown in Table 1. As a resul t of this, three standard vehicles (I, II, and HI) have been established which a re representat ive of the LFV, LEM,

2

and LLS size vehicles . The standardized dimensions a re shown in Table 2. The radius of gyration about the bank (roll) axis is assumed equal to that shown for pitch. The radius of gyration in yaw (about the vehicle 's centerl ine) is taken to be 0. 9 t imes the pitch value. Having standardized the three vehicles, all touch-down dynamics calculations performed for Vehicle I can be made applicable through dynamic scaling factors to Vehicle II and Vehicle EI.

Eight vehicles a re considered, as shown in Figure 1, corresponding to four values of the ratio of landing gear diameter to center-of-gravity height. Both three-legged and four-legged vehicles a re considered even though the LFV, LEM, and LLS have all received major emphasis as four-legged vehicles . A major purpose of this study, however, is to obtain es t imates of the relative probability of stable landing with three-legged vehicles compared to four-legged vehicles . Although the d iametere.g . height rat ios for LFV, LEM, and LLS vary from 2. 50 to 3. 25 (Table 1), the values chosen for this generalized study vary from 2.0 to 3 .0 . The large landing gear diameter designed for the LLS was based upon very severe initial conditions which a re believed to be overly conservative, result ing in an excessively heavy landing gear .

The spring ra tes assumed for the vehicles of this study (Fig. 2) a re the equivalent vert ical spring ra tes per leg when landing on a level, high-friction surface with no sliding of the foot pads . These ver t ical spring ra tes were obtained by assuming an elastic spring ra te (7005 N / c m or 4000 l b / in for Vehicle I) along the three s t ruts of an inverted tripodal leg. As seen, the vert ical spring ra te decreases as the landing gear diameter increases , negating somewhat the beneficial resul t of the l a rge r landing gear d iameters . The spring ra tes assumed for the LLS size vehicle (Type III) a re ra ther large and may be very difficult to obtain. These spring ra tes a re required to satisfy dynamic s imi lar i ty . The results, of this study may therefore be somewhat optimistic for this l a rger vehicle because of the high stiffness assumed. On the other hand, the assumed spring ra te for Vehicle I (LFV) is probably low so that the resul ts obtained may be somewhat pess imis t ic . Previous study (ref. 5) has shown that decreasing stiffness has a detrimental effect on landing stability.

The force-s t roke relationship in each strut is shown in Figure 3. Upon compression, each s t rut is assumed to compress elastically until the limiting force in compression is reached. Fur ther stroking takes place with a constant stroking force assumed. As the s t rut begins to re-extend, the stored elastic energy is re leased until the s trut force reaches ze ro . Fur ther re-extension of the s t rut takes place under zero load or f ree - re tu rn conditions. No limiting force in tension is assumed. The same spring constant is assumed for the s trut in tension as used in compression. The limiting force is assumed to represent the crushing force of aluminum honeycomb energy absorbers in the s t ru t s . Upon subsequent stroking, the s t rut shortens under zero load as long as it is still in the f ree - re turn stroking region.

3

The limiting force or crushing force values for the eight vehicles were determined so that the deceleration load factor is limited to 3. 0 earth g's when the vehicle lands on a level, high friction surface on all legs simultaneously. This load factor appears to be a good choice from previous study (ref. 6 ) . If the chosen load factor is too high, a landing gear designed for such loads is too heavy. If the load factor is too low, excessive stroking is required to absorb the landing energy, thus increasing the s t rut length required and landing gear weight. Each leg is considered to be an inverted tripodal arrangement of one main s t rut and two lower support s t ru t s . The limiting force of each support s t rut is considered to be one-third of the main strut limiting force. This relationship is believed to be a good choice based upon previous analysis (ref. 7 ) . The limiting forces obtained for the eight vehicles a re shown in Table 3. These values correspond to the forces for Vehicle I (LFV) . Corresponding forces for the la rger vehicles a re increased by the dynamic s imilar i ty force factors given in Table 4 .

The mass of Vehicle I (LFV) is considered to be 453. 59 kg (earth weight of 4448 N or 1000 lb ) . The mass rat ios given in Table 4 can be used to obtain the mass of the la rger vehicles .

INITIAL CONDITIONS

Initial conditions have been obtained using a Monte Carlo approach which randomly selects values from zero to one and re la tes these values to the initial conditions sought through the distribution functions that have been established for the initial var iables . Determining a reasonable distribution function for each of the variables of in teres t is the most questionable par t of the entire study because of the lack of data. However, one value of this study may be that of helping to point out the need for detailed and real is t ic stat ist ical information concerning initial touchdown conditions.

Various simulations have been performed ( refs . 8-13) which have yielded some information on the conditions at touchdown for lunar landing. All of these simulations have been conducted with varying degrees of r ea l i sm relative to instrumentation and control response, pilot training, tes t objectives, pilot motiva-tion, and other factors . Results must be viewed with some degree of caution since they may not real is t ical ly simulate an actual vehicle with well t rained and motivated pi lots . Future efforts, it is hoped, will provide more real is t ic informa-tion on touchdown conditions through the use of the Lunar Landing Research Facility (ref. 14) and also the free flight Lunar Landing Vehicle (ref. 15).

As far as the vert ical and horizontal components of touchdown velocity a re concerned, it is reasonable to expect that the vast majority of cases of actual landings with well motivated pilots will be very soft landings with low touchdown

4

veloci t ies . The simulations reported by Hill (ref. 9) show that the ver t ical descent ra te is less than 1. 2 m / s (4 fps) for 90 percent of the ca ses . The forward velocity at touchdown was also reported as less than 0. 6 m / s (2 fps) for 90 p e r -cent of the c a s e s . The simulation resu l t s reported by Wood and Post (ref. 12) indicate more severe landing velocit ies. About 35 percent of the t ime the pilots failed to land within the design envelope of 3. 0 m / s (10 fps) vert ical velocity, and 1. 5 m / s (5 fps) horizontal velocity. It was concluded that this large failure ra te can be attributed not only to difficulty of the task, but also to the training level of the pilots and limitations in the simulation. It was believed that grea ter pilot experience, and improvements in displays and display precis ion will reduce such failures considerably in future s tudies.

It is expected that the frequency functions of the vert ical and horizontal velocity components should have some tail-off toward higher velocities to represent more severe c a s e s . The frequency function chosen as a reasonable representat ion of the vert ical and horizontal velocity components is given in Figure 4. This function corresponds to the chi-square frequency function with three degrees of freedom. Appropriate choices for multiplication constants convert the values of the variable x to the vert ical and horizontal velocit ies.

Concerning the lunar slope expected in mar ia regions of the moon, it is expected that the vast majority of landings will be on low sloped t e r r a in with only a small percent of the landings on rougher t e r r a in . It may well be argued that manned landings on rough, highly sloped t e r r a in will never take place. However, for the small LFV exploration system, some landings may well be on s teeper slopes because of the desire to land near interest ing geological and topological fea tures . In addition, if lunar dust somewhat clouds the pi lot 's vision during the final phase of landing, it is possible that one or more pads may impact on a protuberance or into a depression which would increase the effective landing slope. In addition, the pad-surface interaction during impact may well serve to increase the effective landing slope. Experimental evidence ( refs . 16-17) indicates that the bearing strength of the soil is not the sole cr i te r ion for penetration. There is a tendency for the downhill pads to penetrate the soil more deeply than the uphill pads during a landing, and thus increase the effective landing slope. Considering these factors, it is assumed that the frequency function shown in Figure 4 is also a reasonable approximation of the effective slope to be encountered upon landing. Mason and colleagues (ref. 18) have established a s imi lar highly skewed frequency function for a representat ive lunar surface profile from 20 100-kilometer t r a v e r s e s . Their work was performed to establish a baseline for determining lunar roving vehicle requi rements . An appropriate choice of a multiplication factor converts the variable x in Figure 4 into the effective lunar slope.

The distribution (cumulative frequency) function, shown in Figure 5, is used in the Monte Carlo approach to obtain ver t ical and horizontal velocity compo-nents and the effective lunar slope. The values given in Table 5 have been obtained

5

from this distribution function by use of appropriate multiplication constants. The multiplication constants were chosen on the basis of the values emphasized in Table 5. For example, McCauley indicates (ref. 19) that, in the i-iO meter scale, the median slope of the maria is about 5 degrees . Thus, with the assumed distribution, 20 percent of the maria has a slope of about 2 degrees or less , 80 percent is within 10 degrees, and 1 percent is 24 degrees or more .

For the LEM size (Type II) vehicle, it is assumed that 99. 865 percent of the landings will have a vert ical velocity of 3.048 m / s (10 fps) or less , and a horizontal velocity of 1. 524 m / s (5 fps) or l e s s . Thus, with the assumed distribution, 90 percent of the landings have a vert ical velocity of 1. 188 m / s (3.90 fps) or less , and a horizontal velocity of 0.594 m / s (1.95 fps) or l e s s . While the values were based upon the LEM design values of 10 fps and 5 fps (ref. 12), the 90 percenti le values also correspond quite well with the simulation resul ts of Hill (ref. 9 ) . Thus, it is considered that the LEM size vehicle (Type II) will land 50 percent of the time with a vert ical velocity of 0.448 m / s (1.47 fps) or less , with a horizontal velocity of 0. 224 m / s (0. 735 fps) or l e s s .

For the resul ts to be equally applicable to the LFV size (Type I) vehicle and the LLS size (Type III) vehicle, the velocities (also shown in Table 5) were adjusted by dynamic scaling considerations. For example, it is assumed that 90 percent of the landings for the LLS (Type III) vehicle will be with a vert ical velocity of 1.638 m / s (5.374 fps) or l e s s . For the LFV (Type I) vehicle, it is assumed that 90 percent of the landings will occur with a vert ical velocity of 0.706 m / s (2. 32 fps) or l e s s .

The dynamic scaling factors given in Table 4 show that, for dynamic similari ty, the velocity is scaled as the square root of the vehicle 's l inear dimen-sion when landing in the same gravity field. Thus, for la rger vehicles, la rger landing velocities can be tolerated. While this indicates that the la rger vehicles may be eas ie r to land, this table also shows that the spring ra te for dynamic similari ty is proportional to the mass and inversely proportional to the linear dimension when landing in the same gravity field. Thus, since the mass increases to some power of the linear dimension, higher spring ra tes a re required for the la rger vehicles for dynamic s imilar i ty .

The horizontal velocity is assumed to be controlled as far as possible to be ze ro . Therefore, it is considered that the direction of the horizontal velocity vector at touchdown with respect to the principal slope direction has an equal probability of being in any direction. Fur ther study of this problem may show an advantage to having a horizontal velocity of sufficient magnitude to guarantee an uphill landing ra ther than a smal le r residual component that may be in the downhill direction. The assumption of trying to land at zero horizontal velocity, however, should certainly be applicable to unmanned vehicles such as the LLS or an unmanned LEM used as a LEM truck. It is also assumed that the vehicle 's

6

geometric heading orientation with respect to the principal slope direction has an equal probability of being in any direction. This assumption is applicable to unmanned vehicles and also to manned vehicles where vision impairment due to lunar dust might make the determination of the direction of principal slope difficult. Under c lear vision, the pilot would probably orientate the vehicle 's heading to align in a 1-2-i orientation to the slope for improved stability.

The pitch and bank angles at touchdown, referenced to the horizontal, a re assumed to follow a normal distribution with zero mean. Likewise, the vehicle 's angular ra tes in pitch, roll , and yaw a re also assumed to follow the same distr ibu-tion. The normal distribution with zero mean and standard deviation of one is shown in Figure 6. Appropriate multiplication constants convert the x variable to angles and angular r a t e s . The standard deviation for pitch and bank angles is considered to be 2 degrees and is applicable for Vehicles I, II, and III. The standard deviation of pitch, roll , and yaw ra tes is taken to be 1. 60 degrees per second for the Type II (LEM size) vehicle. Pas t simulation studies show that these a re reasonable assumptions. For dynamic s imilar i ty , the corresponding standard deviation of angular ra te is taken to be 2. 70 degrees per second for the Type I (LFV) vehicle and i . 16 degrees pe r second for the Type III (LLS) vehicle.

The 10 variables of vert ical velocity, horizontal velocity, lunar slope, c ross - s lope angle, heading angle, bank angle, pitch angle, pitch ra te , roll rate , and yaw rate a re assumed to be uncorrelated.

The Monte Carlo procedure for obtaining the initial conditions can be summarized in the following manner. A set of values for 400 cases is obtained where each case consists of obtaining 10 pseudo-random numbers from a previously developed digital random number generator subroutine for a rectangular frequency function (ref. 20) . These numbers a re equal to or grea ter than zero and equal to or less than one. These 10 random numbers , RNj through RN10, a r e then used as F (xt) through F (x10) to obtain the values for x t through x10 from Table 6.

This table was obtained from values tabulated by Abramowitz and Stegun (ref. 21) . Linear interpolation is used to obtain the values of x^ to x10. The initial conditions a re then computed:

v , , ^o.oeosxi ht l

v h t = V ^ v h t V = 0 . 1 2 i x

v i

V = 1 / ^ V V Z V

7

(1)

(2)

(3)

(4)

y = 2 .113x 3

CSA = 360 x4

V, = V, x cos CSA h lit

V = - V, , sin CSA z ht

4> o = 360 x5

% = 2 x6

6 = 2 x 7 p

0O = 0 + y u p

w = 2 . 7 x 8 X

co = 2 . 7 x q y 9

wz ^ 2 . 7 x 1 0

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

The values were obtained corresponding to the Type I (LFV) vehicle for convenience in performing touchdown dynamics computations since the time required to obtain the touchdown motion is l e s s . A computing interval of 0. 002 sec was used in performing the touchdown dynamics computations. The same resul ts could have been obtained with the l a rge r vehicles in the. same computation t ime by increasing the computing interval by the appropriate time factors given in Table 4. Tabulation of the initial conditions for the 400 cases is given in Appendix A.

A random number confidence tes t was performed to judge the adequacy of the pseudo-random numbers generated by the digital subroutine. The resul ts a re given in Appendix B. Four thousand random numbers were generated corresponding to 400 cases with 10 random numbers in each case, RNj through RN10, used to obtain the 10 initial conditions. Therefore, 400 values were obtained for each of the 10 random numbers RNi through RN10. The range from zero to one was divided into 10 intervals so that these 400 values would have an expected frequency of 40 in each interval . The observed frequencies were obtained and the chi-square value obtained. These chi-square values should follow a chi-square distribution with 9 degrees of freedom. Therefore, the value of chi -square could be expected to exceed 12.2 (ref. 21) for 20 percent of the cases , or for 2 of the 10 random numbers . None of the random numbers have a chi-square value which exceeds the 12. 2 value. Therefore, the values generated by the random number subroutine a re adequate.

8

TOUCHDOWN DYNAMICS PROGRAM

The touchdown dynamics p rogram used for this study corresponds to a three-dimensional method programmed for digital computer operat ions. The method was developed as a simplified vers ion of the landing dynamics method of analysis for the LEM described by Mantus, Lerner , and Elkins (ref. 22) . This simplified version is res t r ic ted to landings on a plane sloping surface without protuberances or depressions and without sliding of the landing pads along the surface. The surface and body axes orientation used is shown in Figure 7. The velocity c ross -s lope angle is the angle from the principal slope direction to the total horizontal velocity vector . The vehicle 's heading angle defines the orientation of the vehicle 's landing pads. Zero heading angle corresponds to a 1-2-1 pad orientation for a four-legged vehicle and to a i -2 pad orientation for a three-legged vehicle. Heading angles of 45 degrees correspond to a 2-2 pad orientation for the four-legged vehicle. Heading angles of 60 degrees correspond to a 2-1 pad orientation for the three-legged vehicle. These various orientations a re shown in Figure 8.

As previously stated, the initial heading orientation has been taken to have equal probability of being in any direction (360 deg ree s ) . However, as far as the touchdown dynamics p rogram is concerned, 0, 90, 180, 270, and 360 degrees heading orientation is still a i - 2 - i orientation for a four-legged vehicle. Similarly, 0, 120, 240, and 360 degrees heading orientation is still a 1-2 or ienta-tion for a three-legged vehicle. Consequently, the initial heading angle obtained from the Monte Carlo analysis has been converted to an equivalent heading within 45 degrees for the four-legged vehicle and within 60 degrees for the three-legged vehicle for input into the touchdown dynamics p rogram. Thus:

04 ~ 9 0

04 = 4> 0 ~ 9 0

' 04 =

cj) 03 = $ 0 - 240 , 180 <

RESULTS

With the three-legged vehicle having the smal les t landing gear (Vehicle i ) , the 400 cases were reviewed to determine by inspection which cases were more likely to tumble. Most of the cases a re obviously stable because of the small touchdown velocit ies, combined with small surface s lopes. Three of the cases were considered to obviously tumble because of the surface slope being grea ter than 24 degrees . Touchdown dynamics motion was then computed for 45 cases which were believed to have a substantial probability of tumbling. Of these, 14 cases did resul t in tumbling motion.

Based upon these resul t s , touchdown motion was computed for an addi-tional 34 cases which were believed to be stable, but with a substantial degree of uncertainty. Of these, three cases tumbled. Finally, an additional nine cases were chosen which were thought definitely would be stable but lacked the high degree of certainty desired. Of these, none tumbled. Based upon these resul ts , a final review was made of the remaining cases with the conclusion that all would certainly be stable. Therefore, for Vehicle 1, 20 cases of the 400 correspond to tumbling c a s e s .

A summary of the cases used for the touchdown dynamics computations for Vehicle 1 is given in Table 7. It is observed that tumbling resul ted for Case 43 with a lunar slope of only 6. 9 degrees while Case 332, with a lunar slope of 19.2 degrees, was stable. While the lunar slope is a very important pa ramete r to consider in touchdown dynamics analysis , other initial conditions a re also obviously important. Case 43 corresponds to a near 1-2 landing orientation with a relatively high vert ical impact velocity, while Case 332 corresponds to a near 2 - i orientation with a small impact velocity.

Touchdown dynamics motion was then computed using Vehicle 2 for the 20 cases which were previously determined as tumbling c a s e s . All stable cases for Vehicle 1 would also be stable cases for Vehicle 2 since nothing is changed except the landing gear diameter, which is l a rge r . Of these 20 cases , 11 cases tumbled including the 3 cases considered obvious for Vehicle 1.

Touchdown motion was then computed using Vehicle 3 for the 11 cases which tumbled when using Vehicle 2. Of these, four cases tumbled.

Touchdown motion was then computed using Vehicle 4 for the four cases which tumbled when using Vehicle 3. Of these, none tumbled.

A s imi la r analysis was performed for the four-legged vehicles . Vehicle 5 was used and touchdown dynamics was computed for 25 cases which were thought to have a substantial probability of tumbling. Of these, 11 cases tumbled.

11

Computations were then performed for 32 additional cases which were believed to be stable, but with a substantial degree of uncertainty. Of these, three cases tumbled. Finally, an additional 14 cases were considered which were thought definitely would be stable but lacked the high degree of certainty desired. Of these, none tumbled. Three cases (with a slope of over 24 degrees) were considered to be obviously tumbling c a s e s . Therefore, for Vehicle 5, 17 of the 400 cases correspond to tumbling c a s e s .

A summary of the cases used for touchdown dynamics calculations for Vehicle 5 is given in Table 8. No case tumbled corresponding to a lunar slope of less than 15. 5 degrees, while no case was stable with a slope grea te r than 19. 3 degrees . This is a slope spread of only 3.8 degrees compared to a slope spread of 12.3 degrees for the corresponding three-legged vehicle (Vehicle i ) . While 7 cases tumbled for Vehicle i with a lunar slope of less than 15. 5 degrees , only 13 cases tumbled with lunar slopes of 15. 5 degrees or more compared to the 17 cases for Vehicle 5. These resu l t s indicate that a relatively well defined cr i t ical slope can be established for four-legged vehicles compared to three-legged vehicles .

The vehicle with the next larges t diameter (Vehicle 6) was then considered. Touchdown motion was computed for the 17 cases with the resul t that 7 cases tumbled including the 3 cases considered obvious for Vehicle 5. These seven cases were then considered using Vehicle 7 with the resul t that three cases tumbled. Finally, these three cases were considered using Vehicle 8 with the resul t that none tumbled. The cases corresponding to tumbling motion a re summarized in Table 9.

A summary of the sample probabili t ies for stable landing is given in Table 10. The sample probability of stable landing var ies from 0. 950 for the three-legged vehicle with the smal les t landing gear to 1. 000 for the vehicles with the largest landing gear . The lower l imit probability for stable landing with 0. 95 and 0. 995 confidence coefficients for the binomial population has been taken from published tables (ref. 23) based upon the sample size of 400. These resul ts a r e given in Table 11. The table also shows the lower l imit probability for stable landing based upon an approximate equation given by Dalton (ref. 24) . This equation is

P s .(l-C)/enfj

[ (3 + 2 C ) / 4 ] 6 / ( n - n f / 2 ) ,

(25)

where

5 = 0 for n = 0

12

6 = 1 f o r n /= 0 .

This equation is seen to give a very good approximation of the resu l t s for the binomial population.

The lower limit probability for stable landing for the binomial population with a confidence coefficient of 0. 995 is shown in Figure 11. Since neither Vehicle 4 nor Vehicle 8 had any cases which tumbled, touchdown dynamics runs were also made for an additional three-legged vehicle and an additional four-legged vehicle with a d iameter -c . g. height ratio of 2. 75. The three-legged vehicle tumbled for Case 317. This case has the highest lunar slope (28. 2 degrees) of the 400-case sample and the vehicle lands in a near 1-2 orientation. The result ing probability for stable landing for this vehicle is also shown in Figure 11. The four-legged vehicle tumbled for Case 129 and also for Case 317. Case 129 has the second highest lunar slope (27. 3 degrees) of the 400-case sample . The resulting probability for stable landing is also shown in Figure i i .

The probability values for the three- legged vehicles vary in a smooth manner, whereas the values for the four-legged vehicles a re somewhat e r r a t i c . At a d iamete r -c . g. height ratio of 2. 75, the probability of stable landing for the four-legged vehicle is less than for the three- legged vehicle based upon the 400-case sample . It is believed that this resul t would not be obtained if the sample size were increased by an order of magnitude. Similarly, at a d i ame te r -c . g. height ratio of 2. 25, the probability of stable landing for the four-legged vehicle appears to be somewhat high. Consequently, the curve shown in Figure 11 for the four-legged vehicles has been faired in a manner to remove the e r ra t i c nature of the data points.

The resul ts indicate that, for a given probability of stable landing, a three- legged vehicle requires a landing gear diameter only slightly la rger than the diameter required for a four-legged vehicle. Since each leg is folded for flight to the moon and must be deployed before landing, the reliability of landing gear deployment should be grea ter for a three-legged vehicle. With only three landing pads to contact the surface, there is less chance of contacting a protuberance or depression. A three-legged vehicle a s su re s a more positive final res t ing support, whereas a four-legged vehicle may tend to rock back and forth. With only a slightly la rger diameter required, the three-legged vehicle 's landing gear should weigh l e s s .

George C. Marshall Space Flight Center, National Aeronautics and Space Administration,

Huntsville, Alabama, September 30, 1965

13

REFERENCES

i . Deitrick, R. E. and Jones, R. H. ; Surveyor Spacecraft System - Touchdown Dynamics Study, Report SSD-3030R, Hughes Aircraft Company, January 1963.

2. LEM Familiarization Manual; Report LMA 790-i , Grumman Aircraft Engineering Corp . , July 15, 1964.

3. Final Report, A Study of Lunar Flying Vehicles; Report 7217-928001, Bell Aerosystems Company, June 1965.

4. Lunar Logistic System; Report MTP-M-63-1 Vol. I-XI, NASA Marshall Space Flight Center, March 15, 1963. (Confidential)

5. Aero-Astro dynamics Research Review No. 1; NASA TM X-53189, October 1, 1964.

6. Lavender, Robert E . ; Touchdown Dynamics Analysis of Spacecraft for Soft Lunar Landing, NASA TN D-2001, January 1964.

7. Black, Raymond J . ; Quadrupedal Landing Gear Systems for Spacecraft, AIAA J. Spacecraft, Vol. 1. No. 2, 1964.

8. Markson, E . , Bryant, J . , and Bergsten, F . ; Simulation of Manned Lunar Landing, Lunar Missions Meeting, American Rocket Society, July 1962, (ARS Prepr in t 2482-62).

9. Hill, J . A. ; A Piloted Flight Simulation Study to Define the Handling Qualities Requirements for a Lunar Landing Vehicle, Report NA62H-660, North American Aviation, September 13, 1962.

10. Rolls, Stewart L . , and Drinkwater, HI, F red J . ; A Flight Evaluation of Lunar Landing Trajector ies Using a Je t VTOL Tes t Vehicle, NASA TN D-1649, February 1963.

11. Matranga, Gene J . , Washington, Harold P . , Chenoweth, Paul L . , and Young, William R. ; Handling Qualities and Trajectory Requirements for Terminal Lunar Landing as Determined from Analog Simulation, NASA TN D-1921, August 1963.

12. Wood, F . E . , and Post, S.; Final Report, Lunar Landing Simulation IIIA, Report LED-570-4, Grumman Aircraft Engineering Corp . , April 1, 1964.

14

REFERENCES (Continued)

13. Hatch, Howard G., J r . , Algranti, J o s e p h s . , Mallick, Donald L . , and Ream, Harold E . ; Crew Performance During Real-Time Lunar Mission Simulation, NASA TN D-2447, September 1964.

14. Jaquet, Byron M. ; Simulator Studies of Space and Lunar Landing Techniques, Fifth Annual Lectures in Aerospace Medicine, USAF School of Aerospace Medicine, February 3-7, 1964.

15. Feasibil i ty Study for a Lunar Landing Flight Research Vehicle, Report No. 7161-950001, Bell Aerosystems Company, March 8, 1962.

16. Final Report of Lunar Landing Dynamics Systems Investigation, Report MM-64-8, Bendix Products Aerospace Division, October 1964.

17. Final Report; Lunar Landing Dynamics Specific Systems Engineering Studies, Report MM-65-4, Bendix Products Aerospace Division, June 1965.

18. Mason, R. L. , McCombs, W. M. , and Cramblit , D. C. ; Engineering Lunar Model Surface (ELMS), Report TR-83-D, Kennedy Space Center, September 4, 1964.

19. McCauley, John F . ; A Prel iminary Report on the Te r ra in Analysis of the Lunar Equatorial Belt, Report Unnumbered, Geological Survey, U. S. Department of Interior, July 1, 1964.

20. Lester , Roy C. ; Probability of S-IVB/lU Lunar Impact Due to Guidance E r r o r s at Translunar Injection, NASA TM X-53134, September 21, 1964. (Confidential)

21 . Abramowitz, M., and Stegun, I. A. (edi tors ) ; Handbook of Mathematical Functions, Applied Mathematics Series (AMS) 55, National Bureau of Standards, June 1964.

22. Mantus, M. , Lernef, E . , and Elkins, W. ; Landing Dynamics of the Lunar Excursion Module - Method of Analysis, Report LED-520-6, Grumman Aircraft Engineering Corporation, March 6, 1964.

23. Lipow, M. ; Tables of Binomial Upper Confidence Limits on Probability of Fai lure , Report 6120-0008-MU-000, Space Technology Laboratories , July 1962.

15

REFERENCES (Concluded)

24. Dalton, Charles C.; Estimation of Tolerance Limits for Meteoroid Hazard to Space Vehicles 100-500 Kilometers Above the Surface of the Earth, NASA TN D-1996, February 1964.

16

Vehicle I, N=3, D/L=2.0

E ^ Vehicle 5, N=4, D/L = 2.0

3Z Vehicle 2, N=3,D/L=2.25 Vehicle 6, N=4, D/L=2.25

Vehicle 3, N=3, D/L=2.5 Vehicle 7, N=4, D/L = 2.5

3C Vehicle 4, Ns3, D/Ls3.0 Vehicle 8, Ns4, D/L3.0

FIGURE 1. VEHICLE CONFIGURATIONS

Vertical Spring Constant Per Leg8 kv, N/cm x 10 -8

20

15

Vehicle I

2.00 O OK Q K f l Q 7 K ! AA

Diameter - CG Height Ratio

FIGURE 2. SPRING CONSTANTS FOR VEHICLES I, II, AND III

Stro

ke

(Tens

ion)-*m

Forc

e (C

ompre

ssion

) i

/

i -

Force

(Te

nsion

)

Stro

ke

(Comp

ressio

n)

FIG

UR

E 3.

FO

RC

E-ST

RO

KE

REL

ATI

ON

SHIP

ff

i

0.3-

0.2-

0.1-

0-

(

, (X)T"

!

e"

K/2

.. ,

X) f(K}= ' ; W3

L 2

U/2

nf)

B-^

) 2 4 6 8 10 12

X

FIGURE 4. CHI-SQUARE FREQUENCY FUNCTION

F(x)

t

I 0-

I.U

A

O.

U.O

nc.

U.O

A A-

U/r

n 9.

XJ.C

A. ( I 3

V 1 =

3j

? ^ 1

i

.1

i I 1 0

I 1

FIG

UR

E 5.

C

HI-

SQUA

RE

DIS

TRIB

UTI

ON

FU

NC

TIO

N

to

to

F(

1.0-

0.8-

0.6-

0.4-

0.2- n U

-i x

)

X =

0 8o

- =

sl

3-

2-

1 ( ) K

i

2 ; 3

FIG

UR

E 6.

N

ORM

AL

DIS

TRIB

UTI

ON

FU

NC

TIO

N

|CVJ CM

>

>

M

(A M < in

o A

X

M

X O

4> 4. ^ M 0 0

y^ X >- M

w - - X *35 -&-"-e a* o o> o S1 "" p a p J

%$=m k " e^t

< <

^ O C gog 5 .t: a y

23

24

Luna

r Sl

ope,

y,

deg

CSA

=

30

, V v

= 1.8

m

/sec s

Vht =

0.

9 m

/sec

i O

A

-

CV 16-

12- 8- 4- \ 1

$P=^

f 0=C

jJ K=

Ctly

=

C

tJZ=

0

r 3 ' 1 w

V

1 4

p 1 V

A

i

v

Tum

ble ||

A St

able

1

r k

v

A

1 1* ^X

^

*

^

/ / r / '

60

-40

-20

0

20

40

60

Head

ing

Angle

,

4, de

g

FIG

UR

E 9.

ST

AB

ILIT

Y B

OU

ND

ARY

FO

R V

EHIC

LE

3

Luno

r Slo

pe, j %

cleg

i

20-

16-

12- 8- A - 1

L

i

I

y ^S

1

CSA

=

30

, V v

= 1.8

m

/sec s

V^ =

0.9

m

/sec

$p =

fo

=

x

=

wy=

wzs

O

V-

' -X

/i

y u ^

v

\v

^

Tumb

le 1

A Sta

ble

1

^v.

y /

A

S A

-50

-40

-30

-20

-10

0

10

20

30

40

50

Headi

ng

Angle

, 9

, de

g

FIG

UR

E 10

.

STA

BIL

ITY

B

OU

ND

ARY

FO

R V

EHIC

LE

7

Probability of Stable Landing f p

iOO

o w O

.97

9 c

.94

.92

.91

w U i ^ ? J

2.00 2.25 2.50 2.75 3.00 3.25 Diameter - CG Height Rati, D/L

FIGURE 11. LOWER LIMIT PROBABILITY OF STABLE LANDING

27

TABLE 1. GEOMETRIC PROPERTIES OF LFV, LEM, AND LLS

Veh ic le

L F V

L E M

LLS

Veh ic le

L F V

L E M

LLS

D

c m (in)

279 (110)

813 (320)

1966 (774)

D / L

2. 50

2. 54

3. 25

L

c m (in)

112 (44)

320 (126)

605 (238)

Y a / L

0 . 6 6

0. 53

0. 63

Y a

c m (in)

74 (29)

170 (67)

381 (150)

Y p / L

0. 27

0. 37

0. 30

Y r P

c m (in) c m

30 (12) 56

119 (47) 208

180 (71) 330

r / L k z

0 . 50 0.

0. 65 0.

0 . 5 5 0.

TABLE 2. GEOMETRIC PROPERTIES OF VEHICLES I, II,

Veh i c l e

I

II

I I I

Vehic l e

I

II

I I I

Type

L F V

L E M

LLS

Type

L F V

L E M

LLS

Y a / L

0. 61

0. 61

0. 61

Y a c m (in) 68 (27)

Y p / L

0. 31

0. 31

0. 31

Y P c m (in) 35 (14)

195 (77) 99 (39)

369 (145) 188 (74)

r / L

0. 57

0. 57

0. 57

r

c m (in) 64 (25)

182 (72)

345 (136)

kz

(in) c m (in)

(22) 58 (23)

(82) 165 (65)

(130) 259 (102)

/ L

52

52

4 3

AND III

k / L z

0 . 4 8

0 . 4 8

0 .48

k z

c m (in) 54 (21)

153 (60)

290 (114)

28

TABLE 3. LIMITING FORCES FOR TYPE I VEHICLES

Veh ic l e

1

2

3

4

5

6

7

8

N

3

3

3

3

4

4

4

4

D/L,

2. 00

2. 25

2. 50

3 .00

2. 00

2. 25

2. 50

3. 00

C N

3576

3897

4218

4977

2682

2922

3162

3738

m p lb 8 0 4

8 7 6

948

1119

6 0 3

657

711

8 4 0

N 1192

1299

1406

1659

8 9 4

9 7 4

1054

1246

P lb 268

292

316

373

201

219

237

280

N/ cm 8756

7425

6252

4588

8756

7425

6252

4588

l b / i n 5000

4240

3570

2620

5000

4240

3570

2620

TABLE 4. DYNAMIC SCALING FACTORS

Quant i ty

A c c e l e r a t i o n

M a s s

Length

A n g u l a r A c c e l e r a t i o n

T i m e

Angu la r Ve loc i ty

Ang le

Veloc i ty

F o r c e

M a s s Dens i ty

Momen t of I n e r t i a

S t r e s s

Spr ing Ra te

Sca le F a c t o r

g / g j = a

m/m = p

L / L j = y a/y

(y/a) / 2

(a /y) / 2

(a /y)

(a y)l/z

a (3

P/y3

P y2

a p /y 2

a p / y

Vehic

II

1. 00

14 .00

2 . 8 6

0. 35

1. 69

0. 59

1.00

1. 69

14. 00

0. 60

115

1. 71

4 . 9 0

l e Type

III

1. 00

42 . 00

5 .41

0. 18

2. 33

0 . 4 3

1.00

2. 33

42.00

0. 27

1229

1.44

7 .76

29

TABLE 5. INITIAL CONDITIONS FROM CHI-SQUARE DISTRIBUTION

F(x)

X

\ t

V

%

-

-

-

20.

1.

0.

0.

000

005

061

122

50.

2.

0.

0.

000

366

143

286

80.

4.

0.

0.

000

642

281

562

90.

6.

0.

0.

000

251

378

756

95

7.

0.

0.

000

815

47 3

946

99.

11.

0.

1.

000

341

686

372

99.

16.

0.

1.

865

022

970

939

99.

21.

1.

2.

997

000

271

541

y deg 2 . 1 5 . 0 * 9 . 8 1 3 . 2 16. 5 2 4 . 0 33 .9 4 4 . 4

V h t , I m / s 0 .057 0 . 1 3 3 0 .261 0 . 3 5 3 0 .441 0 .637 0 .901 1.181

V, TT m / s 0 . 0 9 6 0 . 2 2 4 0 .442 0 . 5 9 4 0 . 7 4 4 1.078 1.524* 1.996 htj-L-L

V h t , HI m / s - 1 3 3 0 . 310 0 .608 0 .819 1.024 1.486 2 .100 2 .752

VV j I m / s 0 . 1 1 4 0 . 2 6 6 0 . 5 2 2 0 .706 0 . 8 8 2 1.274 1.802 2 .362

V , II m / s 0 . 1 9 2 0 . 4 4 8 0 . 8 8 4 1.188 1.488 2 .156 3 . 0 4 8 * 3 .992

V III m / s 0 . 2 6 6 0 .620 1.216 1.638 2 . 0 4 8 2 .972 4 . 2 0 0 5 .504

*Values u s e d to e s t a b l i s h m u l t i p l i c a t i o n c o n s t a n t s ,

30

TABLE 6. TABULATION OF DISTRIBUTION FUNCTIONS

N o r m a l D i s t r i b u t i o n C h i - S q u a r e D i s t r i b u t i o n (x = o,

TABLE 7. SUMMARY OF TOUCHDOWN DYNAMICS RUNS FOR VEHICLE 1

Group A

C a s e R e s u l t y, deg

6

8

9

14

39

43

46

64

66

82

84

87

118

120

144

148

157

158

173

177

179 187 198

206 207

211

212

214

S

S

S

S

S

T

S

S

S

S

T

T

T

S

S

S

T

S

T

T

S

S

S

S

S

S

T

S

13.0

10. 3

11.8

14.2

15.7

6.9

8.9

12. 4

T2.8

15. 3

15. 5

11. 5

10.7

10. 3

14. 1

11.8

23.0

16.0

15. 5

19. 3

10.4

9.1 14. 9

16.8 16.9 11.3 9.7

11. 3

32

Group B

Case Result y, deg

145 S 10.4

168 S 12.7

172 S 1.7

235 S 10.0

238 S 9.7

242 S 12.4

247 S 8.4

250 S 8.0

259 S 6.8

275 S 13.4

277 S 15.0

288 S 1 2. 4

289 S 8.8

291 S 7.2

296 S 8.6

303 S 9.5

307 S 14.8

309 S 10.7

314 S 13.0

315 S 12.1

319 S 7.3

321 S 13.5

327 T 14.9

338 S 6.9 341 S 8.4

357 S 15.3 363 T 14.0

364 S 15.4

Group C

Case Result y, deg.

13

61

95

100

102

103

123

243

337

S

S

S

S

S

S

S

S

S

14.8

9.5

8.4

14. 3

13.9

4.9

8.0

12.2

8. 3

TABLE 7 (CONTINUED)

G i

C a s e

215

2 3 4

2 9 4

299

3Z8

332

3 3 3

339

3 4 5

346

350

351

371

379

3 8 9

390

398

oup A

R e s u l t

S

S

S

S

S

S

S

T

S

T

T

T

T

S

S

S

T

y, deg .

17 .7

1 4 . 4

1 6 . 4

1 1 . 6

17 .7

19 .2

1 6 . 3

17 .0

17 .2

2 1 . 8

2 0 . 6

16. 5

2 1 . 2

14 .7

1 0 . 9

18. 3

2 3 . 5

Group B

C a s e R e s u l t

368

369

386

396

397

4 0 0

Group A:

Group B:

Group C:

S

T

S

S

S

S

C a s e s

,

y, deg .

14. 1

12. 5

9 . 5

13 . 3

8 . 7

6 . 7

which w e r e thought to h a v e a

s u b s t a n t i a l p r o b a b i l i t y of t u m b l i n g .

C a s e s wh ich w e r e be l i eved , with

s u b s t a n t i a l u n c e r t a i n t y , would be

s t ab le

C a s e s which w e r e thought , with only a

s m a l l d e g r e e of u n c e r t a i n t y , would

be s t a b l e .

33

TABLE 8. SUMMARY OF TOUCHDOWN DYNAMICS RUNS FOR VEHICLE 5

368

371

390

398

S

T

T

T

Gi Case

14

39

66

84

157

173

177

198

206

215

289

296

315

319

328

333

339

346

350

351

364

oup A Result

S

S

S

S

T

T

S

S

T

S

S

S

S

S

T

S

T

T

T

T

S

y. deg.

14.2

15.7

12.8

15.5

23.0

15.5

19.3

14.9

16.8

17.7

8.8

8.6

12. 1

7. 3

17.7

16.3

17.0

21.8

20.6

16. 5

15.4

14. 1

21. 2

18. 3

23.5

Gi Case

6

9

43

46

61

82

95

103

123

144

158

207

212

228

oup B Result

S

S

S

S

S

S

S

S

S

S

S

T

S

S

Y< deg.

13. 0

11.8

6.9

8.9

9.5

15. 3

8.4

4.9

8. 0

14. 1

16.0

16.9

9.7

9.3

Gi

Case

13

64

87

100

102

118

145

148

275

303

321

338

389

396

-oup C Result

S

S

S

S

S

S

S

S

S

S

S

S

S

S

y, deg.

14.8

12.4

11. 5

14. 3

13.9

10.7

10.4

11.8

13.4

9.5

13. 5

6.9

10. 9

13. 3

234

238

270

277

288

291

294

307

314

327

332

337

345

357

363

369

376

379

S

S

S

S

S

S

T

S

S

S

T

S

S

S

S

5

S

S

1 4 . 4

9.7

6.7

15 .0

1 2 . 4

7 . 2

1 6 . 4

14.8 , , Q Group A: C a s e s which w e r e

thought to have a 1 4 . 9 s u b s t a n t i a l p r o b a -, q y ' b i l i ty of t umb l ing

. 3 G r o u p B : C a s e s which w e r e

1 7 . 2

1 5 . 3

14 .0

12. 5

5 . 4

14 .7

b e l i e v e d , with s u b -s t a n t i a l u n c e r t a i n t y , would be s t a b l e

Group C: C a s e s which w e r e thought , with only a s m a l l d e g r e e of u n c e r t a i n t y , would be s t ab l e

TABLE 9. SUMMARY OF CASES WHICH TUMBLE

C a s e Veh ic l e

36

.43

84

87

118

T

T

T

T

T

2

T

T

3

T

5

T

6

T

129

157

173

177

206

207

212

294

317

327

328

332

339

346

350

351

363

369

371

390 398 n

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

20

T

T

T

T

T

T

T

T

T

11

T

T

T

4 0

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

17

T

T

T

T

T

T

7

T

T

T

3 0

35

TABLE 10. SAMPLE PROBABILITY OF STABLE LANDING

V e h i c l e

"Ps

1

0 . 9 5 0 0

p = (400 -s

2 3

0 . 9 7 2 5 0 . 9 9 0 0

n f) / 4 0 0

4 5

1 .0000 0 . 9 5 7 5

6

0 . 9 8 2 5

7 8

0 . 9 9 2 5 1 .0000

TABLE i i . LOWER LIMIT PROBABILITY OF STABLE LANDING

Vehic le

p (Ref 23)

p (Ref 24) s

Veh ic le

P s (Ref 23)

Ps (Ref 24)

C = 0

1

0.9282

0.9303

C = 0.

1

0.9149

0.9223

950

0.

0.

995

0.

0.

2

9549

9575

2

9439

9498

0

0

0 .

0.

3

9773

9787

3

9689

9713

0

0

0.

0.

4

9925

9925

4

9868

9868

5

0.9369

0.9394

5

0.9243

0.9314

6

0.9674

0.9696

6

0.9577

0.9620

7

0.9807

0.9817

7

0.9728

0.9744

8

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APPEND IX A

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