Chapter 4

Integrated Algorithm for Impact Lunar Transfer Trajectories

using Pseudo state Technique

Abstract A new integrated algorithm to generate the design of one-way direct transfer

trajectory for moon missions is presented. The departure translunar state together with some

unknown Earth parking orbit parameters is determined. The transfer trajectories are generated

including the gravity fields of the Earth and the moon based on pseudo state technique.

Because the main aim is to develop a methodology for determining an initial departure state of

a lunar transfer trajectory, for simplicity, a rectilinear hyperbola that results in impact is

assumed for arrival phase. The translunar injection state, that is the initial state of the transfer

trajectory, is obtained by iteration on the argument of latitude of the parking orbit through

Lambert problem solutions. Lambert solution is obtained using the universal algorithm

presented in Chapter 3. The numerical results obtained using the integrated algorithm are

analyzed and a comparison with the Lambert conic solutions is made. The role of pseudo

state solution in reducing the error in achieving the target is discussed. The reduction in the

error in achieving the target position is found to be more than 95%.

4.1 Introduction

In lunar and interplanetary travel, the space vehicle passes through multiple gravity

fields, which makes the trajectory design more complex. For a lunar mission, the space

vehicle passes through the gravity fields of the Earth and the moon; the trajectory design

problem becomes acomplex 3- body problem. Further, the non-spherical gravity fields of the

50

Earth and the moon also influence the transfer trajectory design. It is well known that there is

no known closed form solution to this problem. The only method that generates near-exact

solution for the 3-body equations of motion is a numerical one. This method is expensive and

computationally intensive as it involves exorbitant number of simulations. Also the

convergence depends on the initial guess made on the initial state. Evidently, this method is

not suitable for mission planning purposes. As an altemate, analytical methods based on point

conic, patched conic and pseudo conic techniques can be used for quick mission design and

analysis. These techniques differ on the force model used in the transfer trajectory design

process: the point conic technique ignores the gravity field of the moon and considers only

the Earth; the patched conic technique considers one body at a time; the pseudo state

considers the Earth for full flight duration and superimposes the moon's effect. Unlike in the

interplanetary transfers, for lunar transfer a different approach is required because the gravity

fields of the Earth and the moon overlap. As described earlier, the transfer trajectory design

process consists of two parts: (i) determining the initial departure state (ii) solving the equation

of motion of the spacecraft. Further, only some of the Earth parking orbit parameters are

assumed and other parameters must be appropriately determined. In this chapter, an

integrated algorithm that generates the initial departure state and the unknown parking orbit

parameters concurrently is developed. Appropriate departure state for the lunar transfer

trajectory is generated using a new iterative procedure. The solution of the equation of motion,

that is trajectory propagation is obtained using pseudo state theory. The necessity of such an

algorithm is discussed. An overview of evolution of the pseudo state theory and its application

in the context of interplanetary missions is provided. A comparison with the Lambert's conic

method (point conic method) is also made to establish the accuracy levels obtained with the

integrated algorithm.

51

4.2 Interplanetary and Lunar Transfer Trajectory Phases

When the design of transfer trajectory is attempted using analytical techniques, the

design process is split into several phases to suit the assumptions made in the modeling of

forces acting on the spacecraft. The interplanetary trajectory design involves three major

phases in general: departure hyperbolic trajectory phase relative to the departure planet from

..,..--""MSI /

~i-f 'Plane1" --

Figure 4.1 Interplanetary transfer trajectory phases

52

Figure 4.2 Lunar transfer trajectory phases

the parking orbit to the mean sphere of influence (MSI) of the departure planet (phase 1),

interplanetary transfer trajectory phase relative to the central body (Sun) from the MSI of

departure planet to the MSI of the target planet (phase 2), and approach hyperbolic trajectory

phase relative to the target planet from the MSI of target planet to the parking orbit around the

53

target planet (phase 3). These three phases are to be synchronized to realize an integrated

mission design. Firstly, the interplanetary phase (phase 2) is obtained by treating the target

planets as point masses and by solving the resulting two-body Lambert problem. Then the

determined initial conditions of the transfer trajectory, assumed to be at the MSI of the

departure planet, are transformed into the conditions relative to the departure planet. These

conditions are asymptotic relative to the departure planet. Departure phase is designed to

achieve these asymptotic conditions by proper choice of the parking orbit characteristics.

Similarly, approach trajectories are obtained by selecting an appropriate aim point satisfying

the arrival constraints. A large number of iterations must be carried out on these three phases

. to integrate them.

For lunar missions, since the gravity field of the moon lies within the gravity field of the

Earth, only two phases are required to be designed for the lunar transfer trajectory (LTT). The

departure phase and the cruise phase are combined in to a single phase: from the Earth

parking orbit to the MSI of the moon. The second phase is selenocentric hyperbolic phase :

from the MSI of the moon to the lunar parking orbit (LPG). Appropriate Earth parking orbit

(EPG) characteristics are determined to achieve the combined phase. The state on the

parking orbit, at which translunar injection that puts the craft on course towards the moon is

executed, must be determined together with the impulse requirement for translunar injection.

4.3 Pseudo state Theory and its Evolution

Several analytical methods exist for the trajectory propagation and they can be

grouped, mainly, into three categories (i) Point conic based methods (ii) Patched conic based

methods (iii) Pseudo-state-theory based methods. Point conic and patched conic methods

ignore the presence of multiple gravity fields or consider them one at a time to design the

54

transfer trajectory through the sphere of influence concepts. The relative velocity vectors are

matched at the patch points on the sphere of influence by iteration, in the patched conic

method. The initial state obtained through these methods, when numerically propagated under

realistic force model, deviate from the actual transfer trajectory and fails to meet the desired

mission objectives. The pseudo state theory solves the problem by correcting the major

portion of the error occurring due to patched / point conic approaches without actually going

for numerical integration.

The evolution of the pseudo state theory started with Wilson Jr.'s paper [23], in 1970.

He explained the theory behind the innovative idea in his paper. As described in his paper, the

pseudo state theory to study the motion of a space vehicle under the gravitational influences

of two bodies (termed as primary body and secondary body) is outlined as:

(1) Given an initial state, propagate the trajectory to the desired time (at which the state

of a space vehicle is to be found under the influence of two bodies) as a conic

relative to the primary body.

(2) Transform this primary-eentered state to astate with respect to the secondary body.

(3) Propagate back along astraight line with the velocity vector with respect to the

secondary body to the initial time to find the position of the space vehicle that

would have been in the absence of the primary body gravity field.

(4) Propagate forward to the desired time as aconic relative to the secondary body.

The new state thus obtained is equivalent to the state, which would have been obtained by

simulating the trajectory numerically from the initial state under the influence of the gravity

fields of both the bodies. When the step sizes are small, this method equals the numerical

integration. Byrnes and Hooper [26] developed a multi conic method, a rapid propagation

method that was used in the design of trajectories for the Apollo mission.

55

Designing the trajectory in the presenceeof the gravity fields of the two target planets'. - - .can be considered as a three-body Lambert problem. The above multi step propagation

technique was modified into a one step method by Byrnes [24] to solve the three-body

Lambert problem. In this technique the three-body Lambert problem is converted into many

two-body Lambert problems of connecting two states viz., a departure state and a pseudo

state (corresponding to a target aim point) in a fixed time. Consideration of the pseudo state

instead of the real state of the target body accounts for the influence of the target body.

Unlike in the multi conic method, the sweep back propagation time is not the same as that of

the forward propagation time but a fixed time is chosen during which the secondary body

gravity field is assumed to be active in addition to the central force field. This vicinity within

which the secondary body is assumed to be active was called as pseudo state transformation

sphere by Wilson and later as pseudosphere by other authors. This pseudosphere is a

secondary body-centric sphere with sufficiently large radius. A discussion on the size the

pseudosphere is included in a subsequent section. Essentially, for a lunar trajectory the

influence of the Earth is considered for the total flight duration and the influence of the moon is

superimposed for a fraction of flight duration. The approach trajectory to the target aim point,

generally a hyperbola within the pseudosphere, is obtained by iteration together with iteration

for the transfer trajectory. The technique of Byrnes is more suitable to handle gravity assist

trajectories around the secondary body and it must be modified to just flyby the secondary

body viz., one-way transfer meeting some target constraints.

The target constraints on arrival scenario create additional complexity of fixing an aim

point and an appropriate approach trajectory in the solution process. To avoid this

complication Sergeyevsky et al. [27] introduced a simplification on the approach trajectory by

assuming that it is a rectilinear hyperbola. Because the periapsis of a rectilinear hyperbola is

56

the center of the secondary body and the eccentricity is one, the target aim point gets fixed as

the center of the target body. This partial knowledge about the approach trajectory avoids the

iteration required to obtain hyperbolic approach trajectory characteristics. He has presented

an algorithm to generate transfer trajectories for interplanetary missions. In addition to its

simplicity due to the assumption on the approach trajectory, this method improves the initial

conditions and, hence, the transfer trajectory by correcting the major portion of errors of the

point conic method. It offers better initial guesses for precision-trajectory-generation process

by reducing the number of iterations for convergence. Because the algorithm is presented as

applicable to. interplanetary transfer trajectories, the requirements of the parking orbit

conditions such as the ascending node of the injection plane and the location of trans

planetary injection to attain the transfer trajectory design are not discussed. In general, this

problem is tackled separately in interplanetary missions as it offers such flexibility.

4.4 Lunar Transfer Trajectory Design

In the design of lunar transfer trajectories the flexibility of interplanetary trajectory

discussed in the previous section is not available because the moon (secondary body) lies

within the gravity field of the Earth (primary body) itself. In the case of interplanetary

missions, the orbit transfer is carried out between the orbits of the planets, and hence, the

initial position of the space vehicle is the same as the position of the Earth which can be

obtained from ephemeris data for the given departure time, and the parking orbit requirements

are computed based on departure asymptotic conditions that are represented by excess

velocity vector, V~. Battin[40] presents a mathematical formulation to obtain circular parking

orbit characteristics that achieves the departure asymptotic conditions ( the right ascension

57

and declination of V~ ). The information about the turn angle (e~) that depends on the shape

of the departure hyperbolic trajectory is used to locate the trans planetary injection point. A

strategy that finds the transplanetary injection point from an elliptic parking orbit is discussed

by Bell et al. [21]. The strategy searches along the parking orbit for a location minimizing the

incremental velocity required for achieving the asymptotic conditions.

For a lunar mission, the transfer is carried out between the parking orbit and the orbit

of the moon. Only some of the orbit parameters such as semi major axis, eccentricity and

inclination of the parking orbit is known. Other parameters such as argument of perigee, right

ascension of ascending node are unknown. So, the initial state on the parking orbit is

unknown, and must be found along with the transfer trajectory design by iteration. The

absence of departure asymptotic conditions in the trajectory design of a lunar mission

necessitates a different treatment for generating the translunar injection state. The treatment

must also meet the requirement that the parking orbit plane should contain the target point

(the state of the moon) at the time of injection. In the trajectory design process, appropriate

parking orbit orientation must be chosen. After fixing the plane orientation, the location on the

parking orbit at which translunar injection takes place must be determined. Search technique

along the parking orbit could be followed to find the TLI location. But this involves solution of

Lambert problem for each point on the parking orbit resulting in large computational time.

When the parking orbit is elliptical, the search process becomes complicated because the

argument of perigee of the parking orbit is unknown. Battin [40] has presented an algorithm to

arrive at parking orbit requirements for circum lunar trajectories based on patched conic

technique that could possibly be modified for one-way transfer trajectory. The algorithm does

not allow the flight time to exceed the flight time needed for a parabolic conic connecting the

58

positions and, hence, puts a restriction on the type of conic (only elliptic orbits). Newton

iteration is used to arrive at the transfer angle and, hence, the location for translunar injection.

In this chapter, an integrated approach to obtain parking orbit conditions for translunar

injection and the transfer trajectory design is presented together with elaborate algorithmic

details. The analytical algorithm presents a unified approach in finding the location of TLI for

both circular and elliptical parking orbits. The analytical solution of the three-body Lambert

problem is obtained by finding a pseudo state of the moon and then, by solving the two-body

Lambert problem involving the departure state and the pseudo state. The target aim point is

chosen as the center of the moon because the main aim of this chapter is to derive

translunar/earth parking orbit characteristics. The choice of target aim point fixes the approach

trajectory as a rectilinear hyperbola thus avoiding the need for iteration over the approach

hyperbola. The use of universal algorithm for Lambert problem solution helps to design any

type of transfer trajectories viz., circular, elliptical, parabolic, hyperbolic. Two iterations are

involved on two states: one on the departure state and another one on the pseudo state

corresponding to the aim point. Also the information that the injection for lunar journey starts

at the perigee of the transfer trajectory is derived. Although it is redundant for tangential,

horizontal injection where the parking orbit and transfer trajectory lie in the same plane and

the injection takes place in the horizon plane, it will be useful for conducting non-tangential

non-horizontal injection studies that require out-of-plane as well as in-plane maneuvers before

injection.

4.5 Selection of Earth Parking Orbit Orientation

For a tangential translunar injection that minimizes the impulse requirement, the earth

parking orbit plane must contain the moon's position on arrival. So, the orientation of the

59

parking orbit must be chosen appropriately, that is the right ascension of ascending node (Q p

) must be chosen appropriately. However, such orientation exists only when IBM I< ip , where

8M and ip are the declination of the moon on arrival and inclination of the parking orbit

respectively. There are two such orientations one containing the moon in the ascending

phase and the other in the descending phase.

The geometry of these planes is given in Figure 4.3. Use of Napier's formula in the

spherical triangle AMF of Figure 4.3 results in

The second plane is given by

Figure 4.3 Geometry of parking orbit orientation

60

· tangsm[180- (aM - Q )] = .M

P tan lp

where aM is the right ascension of the moon on arrival.

4.6 Procedure for Determination of Translunar Injection State

For circular orbits, a location on the parking orbit is given by argument of latitude. The

translunar injection impulse requirement is minimum when the chosen location is the perigee

of the transfer trajectory. In the case of elliptic parking orbits, the impulse for translunar

injection is minimum and tangential, when the injection takes place at the perigee of the

parking orbit and the argument of perigee of the transfer trajectory is same as that of the

parking orbit. Instead of searching all through the Earth parking orbit to find appropriate

translunar conditions, this algorithm uses the above-mentioned facts. In other words, for

tangential and horizontal injection, the argument of perigee of the transfer trajectory must be

equal to the argument of latitude of the translunar injection point on the parking orbit. Few

iterations on this parameter (OJ/) through a repeated Lambert problem solutions makes it

possible for the algorithm to find the location of TLI on both types of parking orbits viz. circular,

elliptical. The iterative procedure for the location of TLI on the parking orbit is given along with

the integrated algorithm.

4.7 Integrated Impact Algorithm

The steps of the integrated algorithm that produces the unknown parking orbit

characteristics and the transfer trajectory design are as follows:

61

(1) Fix the following inputs: date of departure td , flight duration tf ' a duration during which

the moon's gravity field also acts on the space vehicle !1t referred to as sweep back duration,

semi major axis a p , eccentricity e p , and inclination ip of the parking orbit.

(2) Find the geocentric position of the moon with respect to the Earth's equator on arrival

(td +tf) using lunar ephemeris data, and set the pseudo state vector R~ = RM •

(3) Compute the right ascension and declination of R~ with respect to the Earth's equator.

(4) Compute the right ascension of ascending node of the parking orbit using the equations

given in Section 4.5.

The iterative procedure to find out the location on the parking orbit to initiate translunar

injection starts here.

(5) Assume avalue for true anomaly vp , and guess avalue for argument of perigee wp of

the parking orbit and set up =wp + v p •

(6) Find the position Rp , corresponding to the assumed parking orbit location described by

(7) Solve the Lambert problem connecting two positions Rp and R~ for the flight duration

t f and obtain the transfer orbit characteristics (at let' it J21' wI' vJ [Evidently 0 p =0t'

ip = it , and vt =O. Though this information is known before the start of the iteration, these

parameters are again determined by the solution procedure of Lambert problem thus

becoming verification for the Lambert problem solution procedure.]

(8) Replace w p +vp of the parking orbit by wt

with the other parking orbit characteristics

remaining the same,

62

(9) Find the new position Rp on the parking orbit corresponding to the updated

characteristics

(10) Repeat the steps 7- 9 till the two successive values of (1), differ only by a very small

tolerance.

(11) Find the arrival velocity VA of the space vehicle on the transfer trajectory (Use the

equations in Section 4.8.1).

(12) Find the hyperbolic excess velocity (if ~) as VA - VM where VM is the velocity of the

moon on arrival epoch and compute a~, 5~ of V~ (Use the equations in Section 4.8.2).

The internal iteration to obtain the parking orbit characteristics ends here. The method

consisting of the above steps is based on 'point conic method' or 'Lambert conic method' in

the literature. If the gravity field of the moon is ignored, the transfer trajectory design is arrived

at the end of step (12). The pseudo state technique, which corrects the trajectory by

including the moon's gravity field, is presented in the following steps and the iteration to find

out the pseudo state starts.

(13) Find the radial distance rh and velocity v" along the rectilinear hyperbola at (t f - ~t )

with V~,M, and PM as inputs using the procedure described in Section 4.8.3.

(14) Because, for a rectilinear hyperbola, radial and velocity directions are parallel, find r"

(-J (-J_ _ V~ _ V~

and v" as r" =rh V~ and V h =V h V~ .

(15) Sweep back from (r", vh ) in a straight line with constant velocity vh for a duration of !1t

and compute the pseudo state as r~ =rh - vh !1t in the selenocentric frame and

transform to geometric frame using R~ =RM + r~ .

63

(16) If the pass is not the first one for the current iteration, then proceed to step (3) with new

R~ . Otherwise compute the time of travel /),i from rh to the periapsis of the rectilinear

hyperbola with the velocity as V_ (use the equations given in Section 4.8.4).

(17) Find the time difference £, =1M-M'I. If £, is less than a pre fixed tolerance, the

solution is reached. Otherwise proceed to (18)

(18) Find the correction (arh) to the pseudo state;at

arh = rh - rh old

at £, -£,old

(19) Correct the asymptotic radial distance rh by

if this is the first iteration.

otherwise.

and reset the old values to the current values with an indication that the pass is not the first

one in the current iteration.

(20) Repeat the steps (14) onwards until £, is less than a prefixed tolerance.

In a typical extemal iteratio~, steps 13-20 are executed, and then again 14-16 are executed.

With the transfer of the internal iteration to step 3, this typical external iteration ends. The two

iterations in this procedure: the first one (internal iteration) to obtain the location of TLI on the

parking orbit and second one (external iteration) to obtain the pseudo state, are repeated until

the convergence criteria on £( is satisfied. The process of convergence requires about four to

five iterations on both internal and external iterations.

64

4.8 Computational Procedures for Some Parameters

The procedures used in the computation of some parameters in the algorithm are

compiled in this section.

4.8.1 Arrival velocity of the spacecraft

. Let aI' e"

iI' Q I' (01' VI be the transfer trajectory characteristics at the initial time

(td ). Find the mean motion n l by

n,=t:a,where f.LE is the gravitational constant of the Earth.

The mean anomaly of the spacecraft on arrival after flight duration t f is given by

M A =n,tf ·

Care must be taken to maintain consistency in units. Other orbital characteristics under Kepler

motion do not change. So the position and velocity of the spacecraft on arrival can be found

from aI' e"

i"Q"OJ"M A'

4.8.2 Right ascension and declination of excess velocity vector

The unit excess velocity vector is given by

[cosa~ COSO~j

V~ = sin~~ COSO~ .smo_

So from the components of V~ I the right ascension (a_) and the declination (0_) are

computed.

65

4.8.3 Asymptotic parameters of a rectilinear hyperbola

Given V~ and !1t, the mean motion is found from

v3

nil =---=- where JiM is the gravitational constant of the moonJiM

and the mean anomaly is given by

M" =-n"At.

The eccentric anomaly F is obtained by solving the hyperbolic form of the Kepler's equation:

M" =sinh(F) - F

The asymptotic radial distance and velocity are given by,

r" = Ji~ [cosh(F) - F]V~

and

4.8.4 Duration of travel on a hyperbola

In the integrated algorithm, step (16) requires computation of travel time along the

approach hyperbola. Given V~, r" and JiM , the semi major axis of the approach hyperbola is

given by

and the eccentric anomaly is obtained from

. h(F) - r"v" Ifsm ----~JiM a"

66

and

F =log(sinhF +cosh F)

The corresponding mean anomaly M h is computed and the travel time is given by

4.9 Illustrations and Analysis

The above algorithm has been implemented and the illustrative results with the

analysis are presented in this section. The size of the earth parking orbit is selected as 300

km circular and the inclination is fixed at 45°. A flight duration of 5 days is assumed for the

transfer and the period for departure is chosen as January 2007, covering about one orbital

period of the moon. The departure is characterized by incremental velocity, right ascension of

ascending node of the parking orbit, argument of latitude on the parking orbit for TLI, and the

arrival is by the hyperbolic excess velocity, right ascension and declination of the excess

velocity vector with respect to the lunar equator.

Generally, the accuracy of the transfer trajectory design is linked to the parameter

sweep back duration in the pseudo state technique. Sweep back duration represents the size

of the pseudosphere. The accuracy levels can be improved by handling this parameter

judiciously. Kredron and Sweetser [32] discuss the importance of this parameter. They give

empirical formulas to compute the optimal sweep back duration for interplanetary missions.

But the flight duration for lunar mission being only four to five days, the optimal sweep back

duration can be found by varying the sweep back durations from 0 to 5 days in small steps.

Obviously, the zero sweep back duration corresponds to the design by point conic method.

The transfer trajectories obtained by pseudo state technique with different sweep back

67

durations are numerically propagated under a force field consisting of both the Earth and the

moon, and the resulting errors on the end constraint, namely reaching the target aim point in

desired flight duration, are assessed. The variations in the solutions with respect to sweep

back duration are shown in Figures 4.4 and 4.5 and its impact on the end constraints are in

Figures 4.6 and 4.7. These plots correspond to a departure date of 171h January 2007, which

require minimum energy for translunar injection in the selected lunar cycle. These figures

show that even the small differences (less than 2 deg.) in the angles results in huge errors in

the end constraints. The error due to point conic solution that ignores moon's gravity field is

about 5.5 hours in flight duration when the target is reached or 25000 kms in the target

distance after the specified flight duration of 5 days. The inclusion of moon's gravity field in the

trajectory design process improves the solution in satisfying the end constraints. Even an

assumption of a nominal presence of the moon (for about only 0.5 days) reduces the errors in

achieving the target drastically (about 50 %). Both the TLI angles increase almost linearly after

an initial rapid increase. The errors are reduced till an optimal sweep back duration is

reached, and beyond this value the errors tend to increase again. For these departure

conditions, the value is about 75 % of the total flight duration, namely, 3.75 days. A further

analysis for other departure dates with different relative geometry scenario and for other flight

durations has also shown the optimal sweep back duration around this percentage value. It

can safely be concluded that transfer trajectory by pseudo state technique assuming moon's

presence for about 75 % of the total duration yields a better solution and satisfies the end

constraints of the transfer trajectory.

Typical lunar mission characteristics obtained using this algorithm are given in Table

4.1 and comparison is made with the Lambert conic results that are obtained using this

integrated algorithm. The initial state obtained by using pseudo state technique when

68

propagated numerically under the gravity fields of the Earth and the moon for the full flight

duration reaches a minimum distance of 423 km from the target aim point, that is the centre of

the moon, in the specified arrival time. This is equivalent to 1315 km below the surface of the

moon. The incremental velocity requirement does not vary between the techniques, but the

angles vary by about 2 deg. As pointed out already the point conic initial state, when

propagated numerically, results in an error of nearly 5.5 hours in arrival time and the space

vehicle is at a very large distance from the target aim point at the prescribed time. The

differences between the solutions of pseudo state and point conic techniques are plotted in

Figures 4.8 and 4.9 for departure and arrival characteristics respectively. The errors in

incremental velocity at departure are negligible. The errors in the argument of latitude of TLI

fluctuate between -2.0 deg to +1.5 deg. The right ascension of ascending node deviates in the

range [1.3 deg, 4.0 deg], indicating that the pseudo state always moves away from the moon's

orbit as seen from reference x-axis. The deviation is at maximum when the moon is near its

apogee and minimum when the moon is near perigee on arrival. It is also observed that the

absolute deviation in 'u ' is maximum when the moon is near Earth equator and the deviation

is near zero when the moon is near its maximum absolute declination. The impact of moon's

gravity field on the arrival excess velocity is about 8 m/s and the variation in the angles, which

fixes the direction of arrival excess velocity, is only marginal.

4.10 Conclusions

An integrated algorithm to obtain the transfer trajectory design together with the

required parking orbit characteristics is presented. Because the main aim here is to develop

an algorithm and related theories for the departure state of the transfer trajectory design, the

aim point is fixed as the centre of the moon. This algorijhm provides better data compared to

69

point conic methods forming a good basis for preliminary lunar mission design analysis with

minimum computational effort. The illustrative results are presented and analyzed. The

accuracy levels of the algorithm are demonstrated. The importance of the sweep back

duration that represents the moon-centered pseudosphere within which the gravity

acceleration of the moon is accounted is highlighted.

70

Table 4.1

Typical lunar transfer trajectory characteristics and comparison

ParameterIntegrated Algorithm using Integrated algorithm using

pseudo state technique point conic method

V, (km/s) 10.8285 10.8281

L\ V (km/s) 3.10278 3.10236

Q p = Q, (deg) 351.172 349.817

u p = OJ, (deg) 168.184 166.576

a, (km) 188242.060 187437.809

e, 0.96452 0.96437

V_ (km/s) 0.89250 0.88830

a_ (deg) -102.526 -102.524

8_ (deg) -1.331 -1.269

Closest Approach Time 22-01-2007 21-01-200700:00:00 hrs 18:39 hrs

Closest Approach -1315.08 -1482.328Distance (km)

Distance from themoon's surface on

-1315.08 23691.3222-01-2007

Ohr: Om: Osec (km)

Departure Date: 17 /01 /2007 0 hrs 0 min 0 sec

71

168.5

168.3

168.1-C)

-8 167.9-GJ

-g 167.7:='S:: 167.5oCE167.3~C)

~ 167.1

166.9

166.7

,.~-

~~

/V

/V

/V

JV

II

166.50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Sweep Back Duration (days)

Figure 4.4 TLilocation variation with respect to the sweep back duration

72

.-""""---.-'~

~

~/

~v

//

1//

II

351.3

351.2-~351.1"0- 351CIJ"0g350.9

g'350.8"0c 350.7CIJ

~ 350.6«J

'0350.5c.2350.4enii 350.3Co)

~ 350.2EC)350.1

a: 350

349.9

349.80 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Sweep Back Duration (days)5

Figure 4.5 Parking orbit orientations with respect to the sweep backduration

73

----~~

~~"""

V......

~VV

/V

I

III

1

0.5

o_-0.5en~

:5. -14)

E -1.5+:0

~ -2'i:~ -2.5c,- -3co:; -3.5'S:4) -4C

-4.5

-5

-5.5

-6 o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Sweep Back Duration (days)

Figure 4.6 Deviation in arrival time on reaching the target

74

•

0.25

\\\

\

'"~'" " ---. l,...--V

"""'"oo 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Sweep Back Duration (days)

2.25

X 104

2.5

0.5

~ 1.75c

~2i 1.5

~~1.25

~c 1c.2...!2 0.75~C

e 2.llI:-

Figure 4.7 Deviation from target distance at the specified arrival time

75

312611 16 21Day of Departure (in Jan. 2007)

6

- Impulse (rnIs)--- Right ascension of asc node (deg),-, -~. Argument of latitude (deg)

, .... .~ .. .. , , ......... .. .. , .... I.. ,

..... ,

'\ ,'\ ,

" 1.(''''·''''.,--, ..,. ..

- /--, ....' '., ...

..., .. ..

~ ... ~,,,,,_,,., ' ......--'--.'

6

5.5

5

4.5

4

!<Q 3.5E 3

~ 2.5CIoS 2.5 1.5

g 1

~ 0.5~C 0

-0.5

-1

-1.5

-2

-2.51

Figure 4.8 Differences (Pseudo-Lambert) in the departurecharacteristics

76

~.

,~"'--''''''''' ." ,_.~

"

I ,;

I '.;, ; '.~

," .' ,, ,.. _-

--j-,~, -"---. ,

... , '.,, ,", ,, ,, ,-,--. V_infinity (m/s) 1-, --- Right ascension of V_inf (d ~g)-. , - Declination of V inf (deg) I-", ,

", ;'" I

.... _...•

6

5

4

3III.! 2~

:1:1 1IIIW 0

~.5 -1

8 -2c4J~ -3:l::is -4

-5

-6

-7

-81 6 11 16 21

Day of Departure ~n Jan. 2007)26 31

Figure 4,9 Differences (Pseudo - Lambert) in the arrivalcharacteristics

77