NASA Contractor Report 185172

/

A Trajectory Generation and System

Characterization Model for Cislunar

Low-Thrust Spacecraft

Volume II--Technical Manual

David J. Korsmeyer, Elfego Pinon Ill,

Brendan M. O'Connor, and Curt R. Bilby

Large Scale Programs Institute

Austin, Texas

May 1990

Prepared for

Lewis Research Center

Under Grant NAG3-928

National Aeronautics and

Space Administration

(NASA-CR-185].72) A [PAJt_CIQ_,Y GENERATIQN

AN3 SYSTfW ChARACTERIZATIqN MQ_FL FOR

CISLUNAR tOW-T_RuST SPACECRAFT. VQLUME 2:

TECHNICAL MA_;UAL (largp _caJe Pro,jrdms

Inst.) 6!._ r, CSCE 21H G3/zo

NgO-21a0/

TABLE OF CONTENTS

Io

II.

2.1

2.2

III.

3.1

INTRODUCTION ................................................................................................ 1

PROBLEM FORMULATION ........................................................................... 3

Vehicle System Characterization ............................................................. 3

Trajectory Generation .................................................................................. 5

2.2.1 Equinoctial Formulation ................................................................ 6

2.2.2 Restricted Three-Body Formulation ........................................ 10

PROGRAM METHODOLOGY ........................................................................... 16

Vehicle Systems Sizing ................................................................................ 16

3.1.1 System Model Program Flow ...................................................... 18

3.1.2 Propellant Estimation ..................................................................... 21

3.2 Trajectory Generation .................................................................................. 22

3.2.1 Orbital Plane Alignment ............................................................... 24

3.2.2 Midcourse Targeting ........ ............................................................... 30

3.3.3 Capture and Circularization ......................................................... 36

Appendix A ................................................................................................................. 47

Appendix B .................................................................................................................. 55

I. INTRODUCTION

With the formation of the NASA Office of Exploration and the

July 20, 1989 speech by President George Bush initiating a space

exploration planning study, more attention has been focused on the

planning and technologies required to establish a base on the lunar

surface. Many different technologies have been considered to

support such an endeavor with a great deal of attention being given

to transportation systems. Many past studies of various lunar base

strategies have led to the conclusion that a major cost driver of a

base on the moon would be the transportation of the infrastructure

from Earth to the lunar surface. One element of the transportation

system is the transfer of crew and cargo between low-Earth orbit

and lunar orbit and the subsequent return of such payloads. Many

technologies have been applied to the space transportation in order

to reduce the program costs. One such technology that may be

applied to orbital transfer is that of constant, low-thrust propulsion

systems. These systems, generally more efficient than conventional

chemical propulsion systems tend to reduce the fuel requirement by

several orders of magnitude and thus reduce the Earth to orbit

launch requirements which results in a cost savings to a lunar base

program. However, it must be noted that low-thrust propulsion

systems are not applicable to crew transfer as the longer trip times

would expose the crew to the hazards of the Van Allen radiation

belts and long-term degradation of human physiological systems.

Therefore, most studies addressing the low-thrust transfer of

payloads from low-Earth orbit to lunar orbit have been primarily in

unmanned cargo vehicles. This is the primary focus of this study.

The analysis contained herein focuses on both the definition of

the vehicle and its supporting systems in conjunction with the

guidance and control algorithms required to fly the spacecraft from

its departure orbit to its destination orbit. The vehicle definition and

systems characterization portion of this study concentrated on

collecting state-of-the-art information from such institutions as
NASA Lewis Research Center, Los Alamos National Laboratory, the

Auburn Space Power Institute, United States Air Force Space

Division, and the Strategic Defense Initiative Organization in order to

develop sizing algorithms to characterize the most state-of-the-art

spacecraft for assessment of its performance. The trajectory analysis

part of this study is the primary state-of-the-art advancement of

this study as it is a three dimensional analysis formulated in the

restricted three-body problem with perturbations

In the past there has been a number of studies conducted on

the low-thrust spacecraft, but most of these studies have either been

dedicated to system characterization or trajectory analysis with very

few being a combination of the two. Most of the trajectory analyses

performed have been done in in-plane without out-of-plane

thrusting required to change inclination of line of ascending node.

This leads to a good initial characterization of a vehicle, but hardly

lends itself to what is required for a complete, robust conceptual

design and mission plan. Other past studies have also concentrated

on a centric trajectory analysis. That is, a problem formulation in a

single gravitational field with a spacecraft in orbit and spiraling

outward (or inward) until escape (or capture) and then changing to a

different centric coordinate systems for continued analysis. It should

be noted that the Earth-Moon systems does not lend itself to this

type of analysis as the Moon is in close proximity to the Earth's

sphere of influence. Therefore, the problem formulation was

developed in the restricted three-body problem to facilitate a more

accurate simulation of the spacecraft's operation.

Since the trajectory analysis technique chosen here is based on

a numerical integration, the vehicle must be initially sized before the

trajectory analysis can begin. Therefore, an initial estimation model

for the spacecraft size was developed which then transfers data to

the trajectory analysis model. After the trajectory analysis is

completed, the spacecraft is re-sized to fit the computed trajectory.

This iteration scheme can be repeated until a suitable level of fidelity

is reached.

2
lI. PROBLEM FORMULATION

To facilitate the development of the low-thrust cislunar

spacecraft trajectory determination and sizing model it was

important to segment the total problem into discrete problems that

could be more easily addressed. From a top-level perspective, the

problem was divided into two functional areas: system

characterization, and trajectory generation and guidance. These two

functional areas were further divided to better enable a solution to

determined.

In the case of the system characterization portion of the model

it was necessary to determine how multiple technologies for various

components were to be integrated into a complete system. Also of

concern was the determination of the amount of propellant required

by the vehicle. The propellant requirement is needed to enable the

determination of an appropriate initial mass of the spacecraft in the

departure orbit. The value of the spacecraft's initial mass is used to

determine the initial acceleration of the spacecraft and the

subsequent acceleration throughout the trajectory generation.

For the trajectory generation and guidance segment of the

model, two reference systems were chosen to allow the greatest ease

of numerical integration, conceptual understanding, and control for

various portions of the spacecraft's flight. The first reference system

is a non-dimensionalized geocentric frame in which six orbital

elements describe the shape and nature of the spacecraft's orbit.

This is the equinoctial coordinate system with modifications. The

second system is a rotating, non-dimensionalized, right handed

coordinate system with the origin at the Earth-Moon system center

of mass. This coordinate system is known as the restricted threebody system. Both reference systems are explained in later sections.

2.1 Vehicle System Characterization

The characterization of a low-thrust orbital transfer vehicle

(OTV) for Earth-Moon transfers was performed by dividing the

3
spacecraft into its major systems. These are power, propulsion,

thermal control, propellant tankage, and structural mass. Each of

these systems is defined in terms of interactions between the other

systems. For example, the propulsion systems needs power, thermal

control, propellant, and structure, the power system needs thermal

control, and structure, the thermal control system needs power, and

structure, etc ....

The payload mass is the primary driver when characterizing

the OTV. The user can input any desired size for the payload and the

system sizing model will characterize a spacecraft based on the

payload mass. The propulsion system is chosen by the user to define

the performance of the system. The propulsion system performance

characteristics, such as specific impulse, and power per engine, along

with the technology choice of the propulsion system are required

user inputs. These inputs allow the model to adequately define the

propulsion system of the OTV. The propulsion system then has

certain requirements that the other systems must provide in order to

have a functioning spacecraft. The technology choice of the power

system is another user input that completes the necessary user

characterization of the OTV. The power system obtains the

information regarding its electrical requirements primarily from the

propulsion system. The thermal control system obtains the required

heat rejection data from the propulsion and power systems and the

structural requirements are developed from the mass and volume

requirements of the propulsion, power, and thermal systems.

Sizing the subsystems required for an OTV is a formidable task

due to the large amount of data needed to cover all of the possible

system choices. In order to implement a computer-based vehicle

systems sizing model, it was determined that the use of parametric

equations would be more effective than using a large design

database to generate system sizing parameters. The use of

parametric equations results in a robust design model. For example,

the parametric equations used in the power system sizing subroutine

were derived by determining mathematical relationships between

4
such system parameters as output power and system mass. Using

these relationships, the power system mass can be calculated for any

desired output power within the range of the power system

technology. All of the system sizing subroutines use parametric

equations, derived from data in current technical publications, to

generate the system mass estimates.

2.2 Trajectory Generation

In previous studies on the development of trajectories for lowthrust cislunar OTVs, little attention has been directed toward the

robust algorithms required for three-dimensional guidance and

control of the spacecraft. The guidance and control of the spacecraft

and the determination of the appropriate trajectory are highly

coupled. The guidance, control, and trajectory determination are

closely related problems which by necessity must be treated with

equal importance 1. A major problem in the design of low-thrust

OTVs and their associated trajectories is the lack of an end-to-end

computer simulation for the spacecraft trajectory between Earth and

lunar orbit. The physical capability of a low-thrust spacecraft to

travel between the Earth and the Moon is known 2. The current

question is how the vehicle will behave at the proposed thrust level

and how it will be guided on its trajectory. To adequately model the

dynamic forces on the spacecraft in its travel in the Earth-Moon

system, two problem formulations were used to generate the

trajectory. The equinoctial formulation of the equations of motion

was used for Earth escape, and the restricted three-body formulation

was used for the for the midcourse and capture phases.

The standard formulation of a spacecraft's orbit is through the

use of classical elements. These correspond to the semi-major axis of

the orbit (a), eccentricity (e), inclination (i), mean anomaly (M),

longitude of periapsis (to), and longitude of the ascending node (f_).

These classical elements are usually defined in terms of a geocentric

equatorial coordinate system with the positive X-axis parallel to a

vector from the sun to the Earth at vernal equinox, and the Z-axis

5
through the north pole of the Earth (see Figure 1). These six

elements are traditionally used to describe the orbit of a spacecraft.

In the classical formulation of the two-body problem, the

angles of mean anomaly (M), eccentric anomaly (E), and true

anomaly (v), are well known and much used. Low-thrust spacecraft

trajectories will always have portions of their trajectories where the

orbit is circular or the eccentricity is very small (<<1). Also, it is

possible to have small to zero inclination orbits when transferring to

geosynchronous orbits or passing through the equatorial plane. The

three angles, M, E, v, are not well suited for small eccentricity or

circular orbits because they are measured from the perigee or

periapsis point of the orbit. When a orbit is circular the periapsis

point of the orbit is not defined. Additionally, co and _ axe illdefined for zero inclination orbits. This causes singularities during

numerical integration of orbits with classical orbital elements. For

these reasons, integration of low-thrust trajectories in classical

orbital elements is not feasible or desirable.

2.2.1 Equinoctial Formulation

An alternate set of orbital elements can eliminate the

singularities experienced by classical orbital elements. These

elements are known an equinoctial elements and are defined in

terms of classical elements,

a=a,

h = e sin (co + f/),

k = e cos (_ + f_),

X=M +o_+f_,

p = tan i/2 sin f2, and

q = tan i/2 cos ft.

The equinoctial element formulation, Pa = (a, h, k, X, p, q), has

no singularities at e=0 or i=0. The semi-major axis, a, remains the

same while h, k, p, and q are just convenient mathematical

formulations, and k is the sum of M, E, and v and is referred to as the

mean longitude.

6
Z

X

Equator

Y

Figure 1 - Geocentric Equatorial Coordinate System

7
The equinoctial coordinate frame is defined by unit vectors f,g,

and w illustrated in Figure 2 and defined as follows.

1 - p2 + q2

A

f= 1 2pq

1 + p2 + q2

-2p

^ 1

g=

1 + p2 + q2

2pq

1 + p2 _ q2

2q

2p

A

w - 1 -2q

1 + p2 + q2

1 p2 _ q2

Where each column vector in brackets contains the X, Y, and Z

A

component of the unit vectors f, g, and w.

The trajectory of a low-thrust OTV can be integrated in this

formulation through the use of a method known as variation of

parameters. A complete derivation of this method is included in

Appendix A. The equinoctial elements are integrated using the

variation of parameters equation written in equinoctial form:

Op ---

]_ _ ¢t "F, 0t

_x {1}

where the fight hand side of {1} is the dot product of the partials

with respect to velocity with the rectangular components of the

perturbing acceleration F. For the low-thrust spacecraft, the

perturbing acceleration vector, F, consists of the acceleration due to

the spacecraft's propulsion system, the perturbing acceleration of the

Moon, the gravitational pull due to the oblateness of the Earth, and

the acceleration on the spacecraft due to atmospheric drag.

The equinoctial elements change slowly during integration and

are smoothly varying functions. This allows the numerical integrator

to take large step sizes, thereby speeding calculation of the

8
^

W

Z

Spacecraft's

OrbitalPlane

f_ Spaccclaft's

Position

Y

Equatorial Plane

Figure 2 - Equinoctial Coordinate Frame

9
trajectory. For both the increase in integration speed and the

elimination of singularities, the equinoctial fornmlation is used to

generate the trajectory during Earth escape.

2.2.2 Restricted Three-Body Formulation

To adequately understand the dynamics of motion of the lowthrust spacecraft, the gravitational effects of the Earth and the Moon

on the spacecraft must be included for the full duration of the

trajectory. The thrusting acceleration of low-thrust OTVs in high

Earth orbit is the same order of magnitude as the perturbing force

due to the Moon. To model the Earth-Moon system with the

necessary accuracy and achieve computational efficiency, the

restricted three-body formulation of the dynamical equations is

utilized as the governing equations of motion.

The problem of three bodies was first formulated in 1772 by

Lagrange. Further studies by Poincare, Laplace, Hill and Szebehely

have resulted in a detailed treatment of the problem and a general

understanding of the interactions between the two primary

gravitational fields. Various formulations are available to represent

the problem of three bodies. Discussions with Victor Szebehely at

The University of Texas at Austin led the authors to use the nondimensional formulation of the restricted problem of three-bodies

for the cislunar transfer and capture portion of the trajectory

generation 3. This formulation has several advantages over the

general three-body problem. The order of equations to be integrated

are reduced from 18th order for the general problem of three bodies

to 6th order for the restricted problem of three-bodies. This

reduction in order dramatically decreases integration time.

Additionally, the equations of motion are non-dimensionalized by the

Earth-Moon distance and the angular rate of the Earth-Moon system

about the systems center of mass (barycenter). The Earth-Moon

system in restricted three-body formulation is shown in Figure 3.

Many realistic orbital cases of interest permit treatment as

restricted three-body situations. A case of particular interest is that

10
Earth

X

Figure 3 - Restricted Three-Body Problem Formulation

11
of a spacecraft moving in the Earth-Moon system. Certain

assumptions can be made about the nature of the Earth-Moon system

that permit a simplified formulation at a slight loss of accuracy. It is

assumed that the motion of both the Earth and Moon is circular and

coplanar about their barycenter. The Earth-Moon line is the x-axis of

a rotating coordinate system and the z-axis is parallel to the angular

momentum vector of the Earth-Moon plane. The y-axis is

perpendicular to both the x-axis and the z-axis. The position of the

Earth and the Moon are constant in the restricted three-body

formulation. The spacecraft, at point P, is assumed to have negligible

mass and to have no impact on the motion of the Earth or the Moon.

The motion of the spacecraft is governed by the relative gravitational

attraction of the Earth and the Moon rotating about the barycenter.

The equations of motion for the spacecraft in the restricted threebody non-dimensionalized rotating coordinate system are:

£- 2y = (2}

_"+ 2J/= Or, and {3}

where /_= __[ 1 (l_u)r12+ #r_ ] + 1-# + /a r 1 r 2 , {5}

r 1 = [(X -//)2 + y2 + 22 ]1/2, and

r a = [(x +1-#)2+ y2+ z2] 1/2 "

A derivation of the equations of motion for the restricted three-body

problem can be found in Victor Szebehely's book, Theory of Orbits 4.

In the restricted three-body formulation of the equations of

motion the Keplerian (potential and kinetic) energy of the spacecraft

is not conserved. This is due to the assumptions regarding the

motion of the Earth and Moon and the spacecraft's effect upon their

motion. However, the sum of the angular momentum, velocity in the

rotating coordinate system, and potential energy of the spacecraft is

conserved. This result is determined through the derivation of the

equation known as the Jacobian integral. The integral and the

12
constant, C, it produces are named after mathematician Karl Gustav

Jacobi who first formulated this integral in 1836. This integral can

be derived from equations {2}, {3}, {4}, and {5}. Initially, equations

{2}, {3}, and {4} are multiplied byx ,y , and z respectively. This

yields,

x" k - 2y" :¢ =//x :¢ ,

_'); + 2._ y = t'/y );, and

_'i= Oz _ •

These three equations are added together to form,

_'J+ j)y+ _i=OxJ+Oyy+Ozi,

which can be integrated and rearranged to find the integral,

C = 2£/- (_ 2 + 3i 2 + i 2). {6}

This constant, C, can be determined for any set of position and

velocity conditions in the restricted three-body problem. If there is

no force acting on the spacecraft other than the Earth and Moon the

value of C, the Jacobian constant, will remain constant. So for a

nonperturbed orbit about the Earth the value of C will remain the

same. For various arbitrary combinations of position and velocity

differing values of C will be found. Conversely, if C is initially

determined to be a particular value from equation {6}, say the value

that corresponds to the position and velocity of a spacecraft in orbit

about the Earth, there will be many combinations of position and

velocity the will give the same value of C. The potential, f_, shown in

equation {5} is dependent solely upon the position of the spacecraft

in the restricted three-body system. If x,y, and z were each set

equal to zero in equation {6} and a value of C was previously

determined from a spacecraft's position and velocity, then equation

{5} will give the x, y, and z coordinates where the spacecraft would

have zero velocity. These coordinates form a surface in the threedimensional coordinate system of the restricted three-body

formulation. This surface bounds the locus of points where the

spacecraft can have motion given the value of its Jacobian constant,

C, in the Earth-Moon system. On this surface a spacecraft with a

13
given C will have zero velocity. Only on the inside of the curve will

the spacecraft have any velocity, thus restricting the motion of the

vehicle to the inside. Figure 4 shows a series of cross sections of zero

velocity surfaces as curves in the Earth-Moon system's x-y plane 5.

The value of the spacecraft's Jacobian constant is used during the

midcourse targeting phase of the trajectory generation. It is used as

an indicator of the spacecraft's ability to achieve the desired cislunar

transfer.

1 Battin, R. H., Miller, J. S., "Trajectories and Guidance Theory for a Continuous

Low-Thrust Lunar Reconnaissance Vehicle," 6th Symposium on Ballistic

Missile and Aerospace Technology, 1961.

2 Hill, P.G., and Peterson, C.R., Mechanics and Thermodynamics of Propulsion,

Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1965,

Chapter 10.

3 Szebehely, Victor, Personal Communication, October 1988.

4 Szebehely, Victor, Theory_ of Orbits, Academic Press, Inc., New York, 1967

5 Kaplan, Marshall H., Modern Spacecraft Dynamics and Control, John Wiley

and Sons, New York, 1976.

14
Earth

C8

X

Ls

Figure 4 - Zero Velocity Curves in the Earth-Moon System

15
|II. PROGRAM METHODOLOGY

3.1 Vehicle Systems Sizing

The vehicle system sizing model contains parametric models of

the major vehicle subsystems: power generation and distribution,

propulsion, support and propellant tankage structure, and thermal

control. In some cases multiple technologies are parameterized for

each system with the technology choice left to the model user. The

interrelationships between these systems are mapped out as a set of

iterating functions that are scaled according to the users system

technology and payload size choice. Figure 5 shows some of the

interrelationships between the various systems. It is assumed that

the OTV does not refuel at the Moon and that no propellant tanks are

discarded. The total dry mass of the OTV is calculated by summing

the system component masses and desired payload mass.

A propellant estimation model has been developed to

determine the appropriate amount of propellant necessary for a

cislunar transfer. A functional relation between the spacecraft's

thrust to weight ratio and the propellant fraction of the vehicle has

been developed using a two-dimensional constant thrust lunar

trajectory program developed by the Large Scale Programs Institute

under a grant from the NASA Johnson Space Center. Using this

functional relationship the propellant mass can be determined which

leads to an initial estimate of the total system mass. The initial total

system mass of the spacecraft, along with the propulsion parameters

(mass flow rate and specific impulse (Isp)), is used as model drivers

by the trajectory generation model.

A description and overall program flow of the vehicle system

sizing model is detailed in the next section. Following the program

flow description, the specific formulation of the propellant estimation

routine is discussed and assumptions concerning the development of

the methodology are detailed.

16
SYMBOLS USED:

m-1 -- payload mass

P-1 - payload power req.

m0 = mass of system

P0 = power requirements

or power output of sys.

T0 = thermal requirements

of system

INPUT: rndot, Isp,

m-l, dV-imp, P-1

mdot Isp

NOTE: System masses

will be used to estimate

mass of structure.

P(etc.)

PROPULSION

SYSTEM MODEL

P(payload)

[m(prop.sys.)]

P(prop.sys.)

.I POWER

-I SYSTEM MODEL

Estimated therm.cont.pow.

requirement

I

i (pow.sys. _

rejection)

T(etc.)

[m(pow.sys.)]

THERMAL CONTROL k

= SYSTEM IMODEL I

[m(therm.cont.)]

T(pow.sys. rejection)

,- P(therm.cont.)

[m(etc.)] "-Sum of system masses

Figure 5 - Basic OTV System Interrelationships

17
3.1.1 System Model Program Flow

The vehicle system sizing model is segmented into input and

systems sizing subroutines. The first subroutine run is the input

subroutine, INPUT0, which checks whether the user wishes to run

the vehicle system sizing model or specify his own overall vehicle

characteristics. If the user wishes to specify the specific vehicle

characteristics, the program prompts the user to enter the initial

spacecraft mass, the total mass flow rate of the propulsion system,

the specific impulse of the propulsion system, and the final mass of

the vehicle. These parameters are transferred to the trajectory

generation subroutines.

If the user chooses to run the vehicle system sizing model, then

the user has the choice of generating the spacecraft's characteristics

for a one-way or two-way trip. A one-way trip would consist of a

Earth to Moon, or Moon to Earth, transfer. A two-way trip is an

Earth-Moon-Earth transfer. At present, even though the system

sizing model can generate characteristics for a vehicle supporting

two-way transfers, the trajectory generation routine only supports

one-way transfers. The characteristics of the spacecraft are output

to a file, LOWTHRST.DAT for later evaluation. After these choices

have been made, program control is passed to the subroutine,

SYSMOD (SYStem MODel), which handles the calls to the various

system sizing subroutines. The top level flow of data is shown in

figure 6.

SYSMOD begins by calling the input subroutine, INPUT1, which

lets the user enter the mass of the payload to be delivered (and

returned in the case of the two-way transfer). Next, SYSMOD calls

the propulsion system sizing subroutine, PROPMOD (PROPulsion

MODel), which allows the user to choose one of four types of electric

propulsion systems. The four types of systems available include ion,

magnetoplasmadynamic (MPD), arcjet, and a user specified system.

There are three options for Ion propulsion: Xenon, Krypton, or Argon

propellant. PROPMOD calls one of the four propulsion system models:

ION, MPD, ARCJET, or USER1. The subroutines, ION, MPD, and ARCJET,

18
Ion

i InputMass Payload

Choose Propulsion System

Arcjet MPD

INPUT

Isp, number of thrusters,

and power input to each

thruster

Size propulsion system based on

user inputs and parametric

equations

I

Choose Power System Technology

!

Size power system and

remaining vehicle systems

using parametric equations

OUTPUT

vehicle system mass breakdown and

system characteristics for user

OUTPUT

mass flow rate, Isp, and

vehicle "wet" mass to

trajectory generation

subroutines

Figure6 -Top-level'Flowof SpacecraftSystem SizingModel

19
allow the user to specify the specific impulse, number of thrusters,

and power input to each thruster. These input quantities are used in

parametric equations to calculate the efficiency of the thruster

system, the dry mass of the propulsion system (i.e. no propellant),

and the total power requirements of the propulsion system. The

subroutine, USER1, allows the user to specify the specific impulse,

mass flow rate per thruster, power input to each thruster, number of

thrusters, efficiency, and the dry mass of the propulsion system.

Once the propulsion system has been specified, control of the

program is returned to SYSMOD which then calls the power system

sizing subroutine, PWRMOD.

PWRMOD allows the user to choose one of six power generation

and conversion systems1, 2. The possible choices include:

(1) Liquid Metal Reactor using Rankine Cycle

Conversion (1.5 kWe 50 MWe),

(2) Liquid Metal Reactor- NERVA Derivative using

Closed Brayton Cycle Conversion

(1.5 kWe - 50 MWe),

(3) Solid Core Reactor using In-Core Thermionic

Conversion (10 kWe 50 MWe),

(4) Liquid Metal Reactor using AMTEC Thermoelectric

Conversion (1 kWe - 50 MWe),

(5) NERVA Derivative Reactor using

Magnetohydrodynamic Conversion

(100 kWe 100 MWe), and

(6) SP-100 reactor with Thermionic conversion

(100 kWe 500 kWe).

Once the power system is chosen, PWRMOD uses parametric

equations to size the reactor, convertor, and control systems based on

the power required. In addition, PWRMOD computes the efficiency of

the power system and returns program control to SYSMOD to call the

thermal control system sizing model, THERM. THERM computes the

mass of the thermal control system based on parametric equations

concerning the efficiency and power output of the power system.

20
The final systems sizing subroutine called by SYSMOD is the

reaction control system sizing subroutine, RCS. RCS uses parametric

equations to estimate the mass of a reaction control system and the

propellant required to give the vehicle an initial estimate of a 100

meter per second velocity change. The vehicle mass used for this

estimate consists of the system masses already calculated, an

estimate of the vehicle's structural mass, and a rough estimate of the

propellant required for the trip. Once this system has been sized, the

only remaining quantity that must be determined is the amount of

propellant required for the trip. The subroutine that contains the

propellant estimation algorithm is called PRPEST.

3.1.2 Propellant Estimation

The subroutine PRPEST (PRoPellant ESTimator) computes an

estimate of the mass of the propellant required for a one-way trip.

The inputs to the subroutine consist of the specific impulse, mass

flow rate per thruster, the number of thrusters in the propulsion

system, the vehicle initial mass (structural mass plus propulsion,

power, thermal control, and RCS system masses), and the payload

mass for the one-way flight. The subroutine uses the mass flow rate

per thruster and number of thrusters to determine the total mass

flow rate for the propulsion system.

PRPEST runs an iterative loop to calculate an estimate of the

propellant mass required for the cislunar flight. The vehicle's

velocity is initialized to zero, the mass of the spacecraft is the

vehicle's initial mass (including the payload mass), and the

propellant mass is zero. The total velocity change, based on Apollo

17 data, required to travel between the Earth and Moon is 9000

meters per second 3 . PRPEST calculates an incremental Av to be

given to the vehicle at each time step in the loop. The equation used

to calculate this Av is:

Av = mdot*g*Isp*dt/m {7}

where,

mdot = mass flow rate (kg/s),

21
g = constant of gravity (km.sec^2),

Isp = specific impulse of the engines (seconds),

dt = time step (100 seconds), and

m = instantaneous spacecraft mass (kg).

The calculated incremental Av from equation 7 at each time step is

added to the vehicle's current velocity to obtain an updated velocity.

The propellant mass is incremented by the total mass flow rate over

the 100 second time increment. The propellant tank mass is

incremented at each time step using a parametric equation that

relates tank mass to propellant mass. The new propellant and tank

masses are added to the vehicle's total mass to obtain a new value

for the vehicle's instantaneous mass. These equations are repeated

until the vehicle's velocity meets or exceeds 9000 m/s.

In order to use PRPEST to estimate the mass of the propellant

required for a two-way trip, SYSMOD first generates the total

spacecraft mass for the "second leg" of the two-way transfer. To

determine the propellant mass for the "first leg" of the transfer, the

propellant mass and tankage is assumed to be part of the first leg's

payload mass. The propellant mass for the first leg is summed with

the propellant mass for the second leg to obtain the total two-way

propellant required.

3.2 Trajectory Generation

The trajectory determination and guidance of the cislunar lowthrust OTV is divided into three distinct phases (see Figure 7):

Orbital Plane Alignment, Midcourse Targeting, and Capture and

Circularization. Orbital Plane Alignment is concerned with the

methodology and required guidance and control to drive the

spacecraft from its initial orbital plane about the Earth into the plane

of motion of the Moon about the Earth. The Midcourse Targeting

phase of the trajectory generation deals with achieving the

spacecraft position and velocity at Earth or Moon escape to achieve

Earth-Moon transfer. The Capture and Circularization phase consists

of the required controls to capture the spacecraft about the target

22
i

¢)

0

° , ,

o Y__

_-.

0

_,,_o

23
planet and to circularize the capture orbit at the desired final

altitude above the surface of the target body. Each of these phases

has a different guidance scheme to achieve the overall goal of

generating trajectories between the Earth and Moon.

3.2.1 Orbital Plane Alignment

The first phase in any cislunar journey for an OTV is the escape

from the initial parking orbit, whether about the Earth or the Moon.

For low-thrust spacecraft to achieve escape, a long period of

continuous thrusting is necessary. This results in a slowly increasing

spiral trajectory from the initial orbit. It can be shown that a nearoptimal thrust for a planar orbital transfer should be directed along

the velocity vector of the spacecraft for the majority of the

trajectory4, 5. This is referred to as tangential thrust, because the

thrusting acceleration will be tangent to the trajectory at all times.

This methodology provides the maximum increase in the semi-major

axis of the spacecraft's orbit over a specific time interval of thrusting.

Tangential thrust is the nominal thrusting approach used in the

spiral escape from the departure planet in this program due to its

near-optimal propellant usage.

The program allows the user to specify the spacecraft in an

orbit about the Earth at any date. To generate a trajectory from the

spacecraft's initial orbit to the final desired orbit about the Moon, the

plane of the spacecraft's orbit must be aligned with that of the

Moon's when the spacecraft is escaping from the Earth. This requires

the spacecraft to transfer between its initial plane of motion to the

Moon's plane of motion. Current low-thrust trajectory research has

not developed adaptive guidance and control algorithms for the

transfer of a spacecraft between two planes using low-thrust

propulsion. This lacking instigated the development of a new

thrusting algorithm for low-thrust transfer between two arbitrary

planes about the Earth.

An arbitrary plane of motion in an Earth centered equatorial

inertial coordinate system can be defined by two of the classical

24
orbital elements, _ and i, of an orbit in that plane. The longitude of

ascending node (_) describes the angle, for a posigrade orbit, from

the x-z plane to the line where the orbital plane and x-y plane

intersect. The inclination (i) describes the angle between the orbital

plane and x-y plane measured at their line of intersection. For any

orbit about the central body of the Earth, _ and i will define the

plane of motion of the object. For two arbitrary planes (see Figure

8), the angles _1 and il define the first plane of motion and the

angles f_2 and i2 define the second plane of motion. The common

angle between the two planes is called the wedge angle, i'. To enable

transfers between two arbitrary orbital planes, Plane 1 and Plane 2,

the angles _1 and il of the spacecraft's orbit, Plane 1, must be driven

to the angles f_2 and i2 of the target orbit, Plane 2. This aligns Plane

1 with Plane 2. In order to change the plane of motion of a

spacecraft it is necessary to determine how low-thrust accelerations

would effect the the angles _ and i that describe the plane.

The influence of perturbing forces upon the classical orbital

elements is known from Lagrange's Planetary equations 6. These

equations address the rates of change of the classical orbital

elements due to perturbing forces on the body in orbit. The

equations that determine effect of perturbing forces on the rate of

change of i and f_ are:

d/ = Fn cos(v + to)

dt V {8}

and

d_ = Fn sin(v + to)

dt V sin(i) , {9}

where Fn is the normal component of the perturbing force, v is the

true anomaly of the orbit, to is the longitude of periapsis of the orbit

and V is the magnitude of the spacecraft's velocity. From equations

{8} and {9} it can be seen that changes in i and f_ are only caused by

a perturbing force acting normal to the spacecrafts orbital plane.

A novel way to determine the necessary thrusting to change

the spacecraft's orbit from one plane to another was developed using

25
s°_ I

I

Plane 1 /

e !

o !

i i

# !

i i

i !

zf i

,t / / /P1_o2

/

I

I

I

I

I

I

I

t

I

Figure 8 - Two Arbilrary Orbital Planes

26
Lagrange's Planetary equations as a starting point. First, it is

necessary to develop a desired di/dt and dt2/dt. The desired average

rates, di/dt and dt2/dt, can be determined by calculating the

difference (Ai) between il and i2 and the difference (At2) between t21

and t22 and developing a time period (At) during which the change

should occur. This At was determined empirically as a function of

the initial acceleration magnitude of the spacecraft. The At is the

minimum time required to achieve escape velocity for a spacecraft,

from starting orbit, under constant tangential thrust. This spiral

escape was calculated in a nonperturbed two body formulation of the

equations of motion (see Appendix B). Then di/dt and dt2/dt can be

approximated by Ai/At and AW/At. The normal force required to

change i and t2 can be derived from {8} and {9} by rearranging the

equations to find,

and

Fni - Ai V

At cos(v + co) {10}

Fnt_ = 6ft V sin(i)

A t sin(v + to) , {11 }

where Fni is the normal force to change il to i2, and Fnta is the normal

force to change _1 to t22.

Since both f2 and i are affected by accelerations normal to the

plane of motion, it is not possible for a low-thrust spacecraft to

change t2 and i independent of each other. Equations { 10} and {11 }

are coupled, so the normal forces derived may be in opposing

directions and effectively cancel during portions of the orbit for a

given Ai/At and At2/At. Some previous studies have attempted to

change only one element, either i or t2, while ignoring the effect the

of the thrust on the other orbital elements 7. These coupled equations

imply that the implementation of the control algorithm that address

only one of the orbital elements makes an incorrect simplification.

Additionally, determining the appropriate control algorithm from

{10} and {11 } would be extremely difficult. An essential

27
simplification, however, has been identified to reduce the two control

equations from {10} and {11} to one single control.

It should be noted that the assumption that the angle between

two arbitrary orbital planes can be closely represented by the

difference between the inclinations, il and i2, is incorrect. The angle

between the two planes, commonly called the wedge angle (i'), is a

function of both f_ and i. The wedge angle is calculated as follows:

i' = cos "1 [c0s(_21 -_22)sin(il)sin(i2) + cos(il)cos(i2)]. {12}

The control developed for use in this program is based on driving the

wedge angle between the two planes to zero. The concepts behind

this control are outlined in the following paragraphs.

The target orbital plane, in the case of an Earth to Moon

trajectory, is the Moon's plane of motion about the Earth. This plane

is defined as the new x-y plane for the geocentric coordinate system.

The x-axis points to the Moon at the time of the spacecraft's escape

from the Earth. The z-axis of this coordinate system is coincident

with the angular momentum vector of the Earth-Moon system. The

classical orbital elements of the spacecraft's orbit are then expressed

with respect to this new frame of reference. Since the desired target

plane, the Moon's orbital plane, is the x-y plane in the new

coordinate system, the spacecraft's inclination in this new system is

the actual angle between the planes or wedge angle. If a thrusting

control is used to drive the new inclination, or wedge angle, i', to

zero, then the f_ of the spacecraft will also change. However, when i'

is zero, f_ becomes undefined. This is because for any orbit in the x-y

plane the longitude of ascending node, f_, is undefined. Thus, the

only control necessary to bring the two planes together is the control

forcing the inclination, or wedge angle, (i'), to zero (Figure 9). This

control is identical to equation {10},

Fni' - Ai' V

At cos(v + tO) , {13}

where Fni' is the normal force required to drive the wedge angle to

zero, v is the true anomaly of the orbit in the new reference plane,

28
Earth - Moon Plane

!

Figure 9 - New Coordinate System for the Simplified Control

29
and co is the longitude of periapsis of the orbit in the new reference

plane.

This control poses practical problems during the integration of

the spacecraft's orbit. If the inclination of the spacecraft's orbit is

zero, or close to zero, integration errors can occur. For this reason the

integration is performed using equinoctial elements. Atmospheric

drag is included in the perturbations to the spacecraft's orbit as is

the oblateness effects of the Earth. These perturbations are added to

that of the Moon for the entire spiral escape and orbital plane

alignment.

When the spacecraft has achieved the required plane change

the problem is essentially reduced to a planar problem. This is

because the spacecraft's plane of motion is now the Earth-Moon

plane and in the absence of out-of-plane perturbations the spacecraft

will stay in the Earth-Moon plane. The planar problem of cislunar

transfer has been previously address to some degree 8. The previous

work has been extended and modified to account for the threedimensionality of the current approach and the modification of the

coordinate system to non-dimensionalized coordinates. The control

of the trajectory generation is passed from the equinoctial

integration subroutine, EQUIN, to the R3BGEN (Restricted 3-Body

GENeration) subroutine which integrates the trajectory in the

restricted three-body problem.

3.2.2 Midcourse Targeting

The midcourse targeting portion of the trajectory generation is

concerned with controlling the spacecraft in order to achieve EarthMoon transfer. The value of the Jacobian constant of a spacecraft is

used as an indicator of sufficient energy for the cislunar transfer.

Figure 10 shows a series of zero velocity curves at various

values of the Jacobian constant. Szebehely notes that when a

spacecraft in the restricted three-body system has a Jacobian

constant of approximately 3.3, an equipotential curve like that shown

at the L2 point in Figure 10 occurs 9. When the spacecraft has an

30
energy level that indicates an associated Jacobian constant of less

than this value, the range of motion of the spacecraft is no longer

restricted only to orbits about the Earth or Moon, but can include

transfer orbits between the neighborhoods of the Earth and Moon.

The midcourse control is based on the desire for the spacecraft

to have slightly under 3.3 for the value of its Jacobian constant when

the spacecraft is in the vicinity of the opening in the zero velocity

curve about the Earth and Moon. This opening occurs at the L2 point

shown in figure 10. The spacecraft needs to have the appropriate

velocity vector in order to pass through the opening between the

Earth and Moon. It is possible, even likely, that the spacecraft will

have enough energy for the opening between the Earth and Moon to

occur, but be in the incorrect position in its orbit to achieve cislunar

transfer. A control is required that will enable the spacecraft to be

travelling in the correct direction to achieve Earth-Moon transfer

when the zero velocity curve opens.

The Jacobian constant of the spacecraft is calculated at each

integration step as the spacecraft nears escape from the initial orbit.

As the value of the spacecraft's Jacobian constant decreases, the

spacecraft's energy in the three-body system (potential, kinetic, and

angular) increases. Figure 11 shows the midcourse control concept.

It is desired that the spacecraft have enough energy at position 1 so

that when the spacecraft reaches position 2 the zero velocity curve

will be open. Different spacecraft acceleration levels correspond to

different rates of change of the spacecraft's Jacobian constant. This

means the spacecraft must be very close to escape energy at position

1 in order to achieve the Jacobian constant of less than 3.3 when the

spacecraft reaches position 2. The midcourse control is based on

determining the spacecraft's energy level or Jacobian constant at

position 1 in order to achieve position 2 using tangential thrust.

Before the midcourse targeting portion of the trajectory is run,

the program presents the user with the option of using a parametric

default value for the Jacobian constant at position 1 or entering a

different number. The default values of the Jacobian constant are

31
Jacobian 4.1

Earth

Figure 10 - Midcourse Targeting Zero Velocity Curves

32
Figure 11- Midcourse Guidance using the Velocity Curves

33
parametrically derived as a function of the spacecraft's acceleration.

This phase of the trajectory generation is user iterative. If the

spacecraft gains too much energy, it will escape the Earth-Moon

system. The user is prompted for a response, after 87 days of

integration time has passed, or if the integration step size falls below

the nominal value. The program gives the user the options of

continuing the trajectory generation or regenerating it with a new

control after Earth escape when either of two conditions is met. On

trajectories from the Moon to the Earth, the same methodology is

used.

The necessary energy for the spacecraft to obtain would ideally

be the value corresponding to the first zero velocity curve that

permits travel between the Earth and Moon. In practice, however,

this is not always the case. The progression of zero velocity curves is

shown in figure 12. The zero velocity curves start about the Earth

and Moon as near circular boundaries, C1. As the energy increases

the two curves meet at a point in space known as the first Lagrange

point, curve C2. At this position, L2, if a spacecraft is placed there

with zero velocity, with respect to the restricted three-body rotating

coordinate system, and there are no perturbations, it will remain

without need for station-keeping. As the energy level increases

further the two curves join to become a single curve, C3, surrounding

both the Earth and Moon. The shape of this curve is similar to the

outline of a figure eight. If the energy level increases slightly

further, the curve, C4, surrounding the Earth-Moon system forms an

opening behind the Moon at L1. This opening means that a

spacecraft with the appropriate value of the Jacobian constant could

leave the Earth-Moon system.

The shape of the zero velocity curves, and the corresponding

values of the Jacobian constant require consideration when

developing guidance algorithms in Earth-Moon space. The difference

between the value of the Jacobian constant where the curves first

join together and the value of the Jacobian constant when the curve

permits escape from the Earth-Moon system is not very large. This

34
C4

Earth

Figure 12 - Zero Velocity Curves in the F_amh-Moon System

35
implies that for Earth to Moon trajectories it would be quite possible

for the spacecraft to obtain a value of the Jacobian constant that

permits escape from the Earth-Moon system and have the

appropriate velocity vector to pass through the opening in the zero

velocity curve behind the Moon. The sensitivity of the trajectory

generation to the midcourse control value (Jacobian constant) during

Earth to Moon trajectories is a direct result of this phenomena. For

Earth to Moon trajectories the difference between the Jacobian

constant for cislunar transfers and the Jacobian constant

corresponding to an open zero velocity curve behind the Earth is

relatively large. Therefore, the sensitivity of the Moon to Earth

trajectory to the midcourse control value is relatively low.

3.3.3 Capture and Circularization

The subroutine CAPTURE controls the capture and

circularization algorithms during the trajectory generation. The

capture algorithm is given preference until the two-body orbital

energy describing a Keplerian orbit about the target planet is

negative. Then the circularization algorithm is used to lower the

eccentricity of the orbit. After the eccentricity is less that 0.1, the

capture algorithm takes over to continue lowering the orbit. If the

eccentricity of the orbit exceeds the 0.1 limit of circularization

control again takes control to lower the eccentricity. The two

algorithms trade control as necessary until the desired final orbit is

reached.

As the vehicle approaches the Moon the capture guidance

phase of the trajectory is initiated. In the absence of impulsive

thrust, the approach to and capture by the target body are critical

and must not necessitate maneuvering beyond the limited

capabilities of the propulsion system. The problem of low-thrust

spacecraft guidance and trajectory determination between the Earth

and the Moon was addressed in a study by Richard H. Battin and

James S. Miller in the late 1950's and early 1960's. The concept for

36
the spacecraft guidance during capture used in this study is derived

from Battin and Miller's work 10.

The guidance scheme is relatively simple and straightforward.

The operation of the capture phase guidance is illustrated in figure

13. The velocity of the spacecraft relative to the target body (i.e., the

Earth or the Moon) is compared with a parametric velocity profile for

a reference spiral capture. The parametric velocity profile is a

function of the radial distance from the target body and the

magnitude of the thrust acceleration. The difference between the

spacecraft's velocity and the parametric velocity profile is used in

combination with the nominal acceleration to determine the direction

and magnitude of the spacecraft's thrust vector during capture.

In order to calculate the parametric velocity profile as a

function of the radial distance from the capturing body the desired

reference trajectory must be generated. The desired reference

trajectory is a spiral trajectory that starts at escape conditions

relative to the target and achieves circular orbit at the desired final

altitude about the target body. The subroutine SPIRAL performs the

generation of the reference trajectory and the calculation of the

parametric velocity profiles. For a detailed discussion of the problem

formulation used to generate the spiral trajectory see Appendix B.

To determine this reference spiral path and the velocity

vectors that accompany it, the spacecraft starts in a circular orbit at

the desired final altitude about the target body. The mass of the

spacecraft at the final altitude is determined by estimating the final

mass of the spacecraft at the completion of the mission. The

spacecraft's trajectory is integrated using negative mass flow from

the propulsion system to spiral out from the target planet using

tangentially directed thrust. The mass of the spacecraft increases

thereby decreasing the acceleration as the integration progresses.

During the generation of the reference trajectory only the

gravitational field of the target planet is considered. The spiral

trajectory is otherwise without perturbations and consequently

remains planar. The calculation of the trajectory continues until the

37
S

a t X[ ac

SI I

Ve _..

ssS SJJS

Pigure 13 - Capture Phase Thrusting Guidance

38
vehicle is on a parabolic trajectory. The associated radial and

tangential components of spacecraft's velocity are recorded at steps

during the integration as functions of the radial distance from the

central body. The velocity components are functions of the radial

distance from the target body. They are obtained by fitting the

recorded velocity components to polynomial and power curves.

Figure 14 and 15 are example graphs of the velocity profiles for

tangential and radial velocity at the radial distance.

A simplified derivation of the thrust guidance control algorithm

is presented as follows. This control algorithm determines the

necessary acceleration vector required for the spacecraft to match a

desired reference trajectory. The actual velocity of the spacecraft,

V v, at a given radial distance, r, is compared with the parameterized

reference capture velocity, V¢, at r. The difference between these

velocity vectors is then determined as V a, where

Vd = Vv - Vc, {14}

For a spacecraft flying on the reference trajectory, the

incremental change in the velocity vector over time, AVe, can be

approximated as the sum of the spacecraft's acceleration vector due

to its thrust and the gravitational attraction of the planet acting over

some small time increment, At. This implies

AVe = (ae + g)At {15}

where ae is nominal acceleration vector of the spacecraft, and g is

the gravitational acceleration vector of the capture planet.

The actual change in the velocity vector of the spacecraft, AVv,

can also be approximated similarly as

AVv = (a t + g) At {16}

where at is the controllable thrust acceleration vector of the

spacecraft.

It can be easily seen from equation {14} that the incremental

change in the difference between the parametric and actual velocity

vectors, AVd, is simply

AVd = AVv - AVc. {17}

39
I¢

F-,

2

0

0

Tangential Velocity .vs. Radial Distance

Vt = 70.7458 ° radA-0.5012 R = 1.00

20000 40000 60000 80000

Radius (kin)

-o- Tang. Velocity

Figure 14 - Tangential Velocity as a function of Radial Distance

4O
-1.36e-20

-1.00e-1

-2,00e-1

-3.00e-1

Radial Velocity .vs. Radial Distance

0 20000 40000 60000 80000

Radius (km)

Radial Vel.

Figure 15 - Radial Velocity as a function of Radial Distance

41
Combining equations {15}, {16}, and {17} and rearranging to solve for

at, the controllable thrust acceleration vector, equation {18} is

obtained,

at = ac - AVa

At - {18}

The thrust acceleration is then chosen so that the rate of

change of the velocity vector Vd is proportional to Vd itself. This

results in

AVd _ Vd

At Tc {19}

where To is an empirically determined time constant. With this

formula the appropriate thrust acceleration can be determined in

both magnitude and direction simply with the knowledge of the

vehicle's position, velocity, and nominal thrust acceleration ac.

In the application of the guidance it is reasonable to assume

that the direction of the thrust acceleration can be varied at will, but

the magnitude of the thrust is limited by the capabilities of the

propulsion system. The spacecraft thrust is never required to

deliver greater than the nominal thrust. The possibility of a

reduction in the thrusting acceleration is not precluded as a desirable

effect of the thrusting algorithm. Figure 16 is a graphical

representation of the acceleration vectors at and ac. The radii of the

circles are determined by the nominal acceleration of the spacecraft.

Then equation {18} becomes,

a t < ac +Vd

Tc.

When the appropriate magnitude of the thrust algorithm is greater

than the nominal capabilities of the engine, a less than nominal

thrust is used in the appropriate direction.

After the spacecraft has been captured by the target body, the

capture algorithm continues to lower the spacecraft's energy level to

bring it in closer to the desired orbit. One distinct difficulty of the

capture algorithm is lowering the eccentricity of the capture orbit.

42
V /T

Figure 16- Thrust Guidance Algorithm

43
For this reason a separate circularization algorithm was developed to

control and lower the spacecraft's eccentricity during capture.

The concept behind the circularization algorithm is very

straightforward. The spacecraft needs to be in a near circular orbit

for the capture algorithm to be most effective. When a orbit is

nearly circular the velocity at each portion of the orbit is

approximately the same magnitude. For eccentric orbits the velocity

of the spacecraft is highest at periapsis and lowest at apoapsis. The

goal of the algorithm then is to drive the spacecraft's velocity to a

uniform value along the entire orbit. At periapsis the spacecraft

needs to increase its velocity, at apoapsis it needs to decrease its

velocity. In order to achieve this, the spacecraft's thrusting vector is

pointed opposite its velocity vector at apoapsis and along the velocity

vector at periapsis. This thrusting behavior translates into the

following controls for the radial atr, and transverse ats, acceleration

components. Then

atr = ac sin(v)

and

ats = ac cos(v)

where v is the true anomaly of the orbit. Figure 17 shows what the

acceleration vector would look like at various points along the orbit.

A potential restriction on nuclear-powered OTVs is the

proposed nuclear safe orbit (NSO) 11. This would be a designated

altitude above the Earth, below which the nuclear powered

spacecraft would be prohibited. The spacecraft would be unable to

descend below the restricted altitude at any point of its trajectory.

This limits the types of trajectories available for cislunar transfer.

1 Advanced Space Analysis Office-Sverdrup/NASA-LERC, "Evaluation of

Advanced Propulsion/Power Concepts," presented to Advanced Space

Propulsion Workshop, April 12-13, 1988.

2 Riehl, J., Mason, L., Gilland, J., Sovey, J., and Bloomfield, H., "Power and

Propulsion Parameters for Nuclear Electric Vehicles," NASA Lewis Research

Center, Space Flight Systems Directorate, Version 1, Release 1, July, 1988.

3 Tsien, H. S., "Takeoff from Satellite Orbits," Journal of the American Rocket

Society, pg 23, July - August, 1953.

44
4 Keaton, Paul W., "Low-Thrust Rocket Trajectories," Los Alamos National

Laboratory Report LA-10625-MS, January, 1986, page 4.

5 Hill, P.G., and Peterson, C.R., Chapter 10.

6 Moulton, Forest Ray, An Introduction to Celestial Mechanics, Dover

Publications, Inc., New York, 1914, 2nd Revised Edition.

7 Bahls, D.L., and Paris, S.W., "A Simplified Method for Obtaining Near

Minimum Time Low Thrust Transfers", AAS 79-166.

8 Korsmeyer, David J., "Trajectory Determination and Characterization of Cislunar Low-Thrust Spacecraft," AIAA/LPI 2nd Symposium on Lunar Bases and

Space Activities of the 21st Century, Houston, 'Ix., April, 1988, LBS-88-080.

9 Szebehely, Victor, Personal Communication, November, 1988.

10 Battin, R. H. and Miller, J.S.

l 1 Galecki, Diane L., and Patterson, Micheal J., "Nuclear Powered Mars Cargo

Transport Mission Utilizing Advanced Ion Propulsion," AIAA/SAE/ASME/ASEE

23rd Joint Propulsion Conference, San Diego, Ca., 1987, AIAA-87-1903.

45
Spacecraft

Periapsis

Point

Figure 17 - Circularization Thrusting Control

46
Appendix A

Variation of Parameters

This is a useful method when a solution is sought to a

homogeneous differential equation under a perturbation. The

homogeneous equation under consideration is the equation of motion

of a spacecraft about the Earth. The equation of motion is commonly

written as,

•- i,tx

xx 3, {A1}

.-o

where Ix is the gravitational parameter of the Earth, x is the position

vector of the spacecraft, and x is the resultant acceleration of the

spacecraft. This equation has a homogeneous solution that derives

the six classical orbital elements aa = (a, e, i, M, co, f2). An alternate,

but equally valid solution derives the equinoctial elements Pa = (a, h,

k, _, p, q). This means the position and velocity of the spacecraft is a

• •

function of time only, so x = x(t), and x = x(t).

If there is a perturbing force, F, to this orbit, the equation of

motion can be written as

_tX -"

x- +F

x 3 , {A2}

this is an associated inhomogeneous differential equation and the

solution to equation {A2} can be found through variation of

parameters.

The basic idea behind variation of parameters is to replace the

constants of the homogeneous solution (the orbital elements) with

time-varying functions. This implies the position is a function of

....4, _ .._

both the orbital elements and time, x = x(pa,t). Then to determine x,

the chain rule for differentiating is used. First the partial of _ with

respect to Pet is multiplied by the partial of Pa with respect to time,

and then the partial or _ with respect to time is taken, so

-:, _x _Pa _x

X-- -I-_

_p_ 3t 3t.

47
However, this equation is constrained by setting

3x _Pa _ 0

_Pa _t , {A3}

this forces the velocity to be the same for the perturbed and

unperturbed case. In effect, this constraint says that the position of

the orbit is not changed at the time the perturbation acts. To

•,

determine the acceleration, x, the chain rule is again used,

3x 3pa 3x

X-- ISPa 8t _t.

,.

However, equation {A1} already gives a solution to x. This implies,

8x SPa _ F

SPa 8t {A4}

The six unknowns of equations {A3} and {A4}, Pa, can be found

by inverting the 6 x 6 matrix of partials. If equations {A3} and {A4}

are put in a matrix, the inverted 6 x 6 matrix of equations would

yield.

If the matrix equation is multiplied through, it is found,

8p --

15 a F

Ox {A5}

where the right hand side of {A5} is the dot product of the partials

with respect to velocity together with the rectangular components of

the perturbing acceleration F. To solve for tia it is first necessary to

know the explicit form of the partial derivatives of the orbital

elements with respect to the velocity components of the reference

orbit. These partial derivatives are obtained by using the Poisson

Brackets 1:

6

_= (pa, _9_' i_1 P13) 8--_ {A6}

48
To use the Poisson Brackets to find the partials of the

equinoctial elements with respect to the velocity, the partial

derivatives of position and velocity vectors with respect to the

equinoctial elements must first be found. The initial step to this is to

determine the position and velocity of the spacecraft from the

equinoctial elements of the orbit. The quantities (X1, Y1, 0) are the

coordinates of the spacecraft relative to the equinoctial frame. These

coordinates must be found and then transformed into the traditional

(x, y, z) of the inertial coordinate system.

It is necessary to recall the coordinate system for equinoctial

elements. The equinoctial coordinate frame is defined by unit

vectors f, g, and w illustrated in Figure A1 and defined by

1 2pq

f-i+p + -2p ,

'Pql _ 1 1 + p2 _ q2

g=l+p2+ 2q ,

w

=

1 +p2+q2

1

I'P

1 -

-2q

p2 .q2

1.

In order to have a variation of parameters program which is

valid for all eccentricities and inclinations it is necessary to use a

formulation which is free of singularities. The key to this is to have

the angles describing the orbit defined for all eccentricities and

inclinations. The equinoctial formulation uses angles called the mean

longitude (_.), and the eccentric longitude (F), rather than the more

classical angles of mean anomaly (M), eccentric anomaly (E), and true

anomaly (v)" {A7}

%=M+o+f_,

49
A

W

Z

Spacccraft's

OrbitalPlane

fl Spacccraft,s

Position

Y

b

v

uatorialPlane

Figu_ A1 - Equinoctial Coordinate Frame

5O
F=E+o_+fl, {A8}

The position in the orbit can be indicated by F. However, in order to

compute the position vector of the spacecraft from the equinoctial

elements, it is necessary to solve Kepler's equation. It is

advantageous to write Kepler's equation in terms of the eccentric

longitude rather than the eccentric anomaly because the eccentric

longitude is defined for all eccentricities and inclinations. Some

elementary manipulations show the Kepler's equation and the

expression for the radial distance, r, from the Earth can be written in

terms of the eccentric longitude,

k = F + h cos F = k sin F, {A9}

r=a[1 -h sin F- k cos F]. {A10}

Kepler's equation {A9} can be solved for F with the standard

Newton-Raphson procedure (or any other iteration method) once the

value of k has been determined from {A7}. Once Kepler's equation

has been solved, the three coordinates (x, y, z) of the inertial system

are obtained with the following matrix equation:

Ix] I Yz = 1+ p2 + q2

1 - p2 + q2 2pq 2p

2pq 1 + p2 _ q2 -2q

-2p 2q 1 - p2 _ q2

Xl

Y1

0

The quantities (Xl, Y1, 0) are the coordinates of the spacecraft

relative to the equinoctial frame. They can be expressed in terms of

F by:

Xl=a[cosF-k+ h(k-F) ]

1 + _/1 + e2J,

Y1 =a[sinF-h+ k(k-F) ]

l+ J.

Also of note are the following time derivations:

F=na

r

{All}

51
h(kcosF+h sinF)

(1 + 3/1" - e 2 ) ],

_'1= na2 [cosF _k(kc°sF+hsinF)]

r (1 + 3/1 - e 2 ) .

The above expressions are valid for all eccentricities and

inclinations. In addition, all the expressions are functions of the

equinoctial rather than the classical elements. Due to the lack of

singularities for circular or zero inclination orbits the equinoctial

formulation is more appropriate for the integration of low-thrust

trajectories.

The partial derivative_

In the variation of parameters derivation it is necessary to

have several partial derivatives of the two-body equations. First, all

of the partials of Xl and Yt with respect to h and k are needed. The

following expressions were taken from work by Broucke2:

_h 1 - _ r ,

a hk133 ] (_.-V) ] -_+l+_sinF(sinF- h13) ,

OY1 a[ (_''F) kh133 ]

- _ - 1 + a cos F (k13 - cos F)

Oh 1 - 13 r , and

-a I(_.- F) (13+ ,203 i_--_1 +_ sin F (cos F- k13)1.

The quantity 13 which has been introduced here is defined as follows:

13= 1

1 + _1 - h 2 - k 2

52
With the use of the above expressions the following partial

derivatives of the _position vector x are easily derived:

0.__x= 1 [_- 3x (t - to)]

ba a 2

ox_OX, +ox,

Oh Oh Oh

-" (:3yI -.

0x OXt _+ g

Ok Ok Ok

Ox _I(1 f+ '_'l

0_.o n

Ox 2

0p 1 + p2 + q2

[q (Y1 f- X1 g)- X1 w]

, and

Ox- 2 [ (X1 -g Y1 Y1 w] --= p - _

0q 1 + pZ + q2

Finally the partials of the equinoctial elements with respect to the

velocity are obtained by using the Poisson Brackets shown in

equation {A5}. The results are then:

0a _ 2x

-=.4,

0x n2a ,

Oq = (1 +pZ+qz) Xl w

Ox 2na25/1 h 2 - k 2

Op _

Ox

(1 +p2+q2) Y1 w

2na2_l - h 2 _ k 2

53
0h k(qYt - pXl)

Ox na2_l h 2 - k 2

_ + _l- h 2- k 2 [(OX1

na 2 L(_-k

Ok _ -h(qY1 - pXl) --w V1

O_ na2_l h 2 - k 2

na 2

Ox na 2 2 na 2 Ok ] Oh

+ 1 [qY1- pXl]

na2_l - h2 - k2

These partials are calculated at each integration time step and

used with equation {A5} to find the time rate of change of the

equinoctial elements due to the perturbing force F. This permits the

integration of the equinoctial orbital elements under perturbation.

1 Broucke, R., and Cefola, P. "The Equinoctial Orbit Elements," JPL Technical

Memorandum 391-238, September 28, 1971.

2 Broucke, R., and Cefola, P. "Numerical Integration of Satellite Orbits with

Equinoctial Elements," JPL Technical Memorandum 391-248, November 5, 1971.

54
Appendix B

Generating Tw0-Dimensi0nal Spiral Trajectories for a Low-Thrust

Spacecraft

The equations of motion of a powered spacecraft in a twodimensional orbit can be derived the spacecraft's kinetic energy,

potential energy, and from the force equation, F = M,a (total force =

mass • acceleration).

If the magnitude of the acceleration of the spacecraft due to

the propulsion system is a, then

W = M,a, {B1 }

where M is the mass of the spacecraft. The spacecraft's acceleration ,

a, is then determined from equations {B1 } to be

a = T._T_

MThe acceleration of the spacecraft due to the thrust of the

propulsion system changes as the total spacecraft mass, M, changes.

As mass is expelled from the spacecraft the total spacecraft mass

decreases. So,

M= Mo - rh,t ,

where t is the time period during which the spacecraft is losing mass

at rh, the mass flow rate of the spacecraft's propulsion system.

To determine the equations of motion of the spacecraft in polar

coordinates the radial and transverse components of the acceleration

due to the thrust need to be calculated from the thrusting force.

Figure B1 shows an arbitrary thrust vector with respect to the radial

and transverse vector. Assuming that the thrust vector makes an

angle ¢ with the local horizon, (measured from the direction of the

motion), the radial, R, and transverse, S, components of the

acceleration are:

a R = !sine, as = rT-T-cos_

M M {B2}

In an unperturbed orbit the only external force present is the

Earth's gravity force which in polar coordinates has the radial

55
V

FigureB1 - Tangential Thrust Components

56
component. The equations of motion in polar coordinates can be

derived quickly from the Lagrangian which is the sum of the kinetic

and potential energy,

L = l (f2 + r 2 62) + GM

2 r

Using the Euler-Lagrange equations, the equations of motion are

obtained as,

• 2 _GM i" r0 =

r 2

and

d(r20) -0.

dt {B3}

If the acceleration components due to the spacecraft's thrust from

equation {B2} are added to the equations of motion {B3}, then

"2

f" - r0 = _GM+Tsin_,

r 2 M

{B4}

d(r20) - r -T- cos_

dt M

The above system {B4} can be replaced by a new system of

four first-order equations. This is done by introducing two new

variables, the radial velocity component u = r, and the transverse

velocity component, aJ = r0. Through the change of variables, {B4} can

ti-_0 - G M+Tsin¢,

r 2 M

d(rv) _ rT cos_

dt M

{B5}

be rewritten as,

The two second order differential equations, {B5}, can be broken into

four first-order coupled equations:

I:=U,

0=32-

r

57
ti = -02 - CLM. + T sin¢_ ,

r r 2 M

--rL(-u + r"rcos#). M

For the numerical integration of these four equations, the state

vector is defined to be (r,O, u, a)),(in this order). These equations

were used in the subroutine SPIRAL to perform generation of the

velocity component profiles for the reference trajectory for spiral

capture. This formulation was especially useful because the

tangential and radial components of the velocity were required to be

save to an array as a function of the radial distance from the planet.

Using this formulation eliminates the need for any cumbersome

calculation of the velocity components and radial distance as the

integration progressed. The necessary information was inherent in

the formulation of the problem.

58
NationalAeronautics and

S#ace Admintstrahon

1. Report No.

NASA CR-185172

4. Title and Subtitle

A Trajectory Generation and System Characterization

Model for Cislunar Low-Thrust Spacecraft

Volume II--Technical Manual

Report Documentation Page

2. Government Accession No. 3. Recipienf's Catalog No.

5. Report Date

May 1990

6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

David J. Korsmeyer, Elfego Pinon III, Brendan M. O'Connor,

and Curt R. Bilby t0.

9. Performing Organization Name and Address

Large Scale Programs Institute

2815 San Gabriel

Austin, Texas 78705

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135-3191

LSPI-903

Work Unit No.

506-64-12

i 1. Contract or Grant No.

NAG3-928

13. Type of Report and Period Covered

Grant Report

14. Sponsoring Agency Code

15. Supplementary Notes

Final Report, Technical Manual. Project Manager, D. Schultz, Mission Assessment and Applications Branch,

NASA Lewis Research Center, Cleveland, Ohio 44135.

16. Abstract

The documentation of the Trajectory Generation and System Characterization Model for the Cislunar Low-Thrust

Spacecraft is presented in Technical and User's Manuals. The system characteristics and trajectories of low-thrust

nuclear electric propulsion spacecraft can be generated through the use of multiple system technology models

coupled with a high-fidelity trajectory generation routine. The Earth to Moon trajectories utilize near Earth orbital

plane alignment, midcourse control dependent upon the spacecraft's Jacobian constant, and capture to target orbit

utilizing velocity matching algorithms. The trajectory generation is performed in a perturbed two-body equinoctial

formulation and the restricted three-body formulation. A single control is determined by the user for the

interactive midcourse portion of the trajectory. The full spacecraft system characteristics and trajectory are

provided as output.

17. Key Words (Suggested by Author(s))

Earth-moon trajectory; Low-thrust; System

characterization; Trajectory model; Low-thrust guidance;

Three dimensional trajectory

18. Distribution Statement

Unclassified- Unlimited

Subject Category 20

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages

Unclassified Unclassified 60

NASAFORM1626OCT86 *For sale by the National Technical Information Service, Springfield, Virginia 22161

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