1

LOW THRUST ORBIT LOW THRUST ORBIT TRANSFER UNDER TRANSFER UNDER

SOLAR ECLIPSE SOLAR ECLIPSE CONSTRAINTCONSTRAINT

M. Guelman, A. Gipsman, A. Kogan, A. Kazarian

Asher Space Research InstituteAsher Space Research InstituteTechnion, Israel Institute of TechnologyTechnion, Israel Institute of Technology

2

Electric Propulsion

Electric propulsion system is a set of components arranged so as to convert electrical power from the spacecraft power system into the kinetic energy of a propellant jet. There are three types of electric propulsion systems :

ELECTROMAGNETICELECTROMAGNETIC

Propellant is accelerated

by the interaction of the

current driven through

the gas with the

magnetic field

ELECTROTHERMALELECTROTHERMAL

Propellant is heated

electrically and then

expanded through

nozzle

ELECTROSTATICELECTROSTATIC

Propellant is ionized

and then accelerated

by a large electrostatic

field

3

Chronology

• 1906 - The earliest known record in which the idea of electric propulsion system for rocket vehicles appears - the notebook of rocket pioneer R. Goddard • 1911 - K. Tsiolkovskii pointed out at a possibility of using the electrical power for acceleration of propellant • 1929 - H. Oberth included a chapter on electric propulsion in his classic book on rocketry and space travel• Late 1950-ies – First experiments on electric propulsion started• July 1964 - First electric propulsion system, SERT I, equipped with ion engines, launched to orbit by NASA• November 1964 - Soviet Zond 2, which carried six Pulsed Plasma Thrusters (PPT) launched to Mars

4

Electric vs. Chemical Propulsion

Chemical propulsion

• Energy stored in chemical bonds of propellant creates thrust - energylimited

ADVANTAGES:• No separate energy

source is required • High thrust to mass ratio

DISADVANTAGES:• Specific impulse is limited

to several hundreds seconds (typically 300-450)

Electric propulsion

• Electrical energy is used to accelerate propellant and create thrust – power limited

ADVANTAGES:• Specific impulse in the

range of thousands seconds

• Low, highly controllable thrust - precise pointing is possible

DISADVANTAGES:• Separate energy source

is required• Low thrust to mass ratio

5

Application of Electric Propulsion

• High specific impulse makes it advantageous option for space missions with large delta V requirement• Low propellant consumption allows to launch more payload, increase the mission duration or to reduce the launch cost• Growth of satellite operation lifetime and availabl e onboard power stimulates the increase of electric propulsion application onboard different spacecraft• At present electric propulsion is used mostly for GEO satellite stationkeeping. • Recent scientific missions that used electric propulsion: Deep Space- 1 (NASA), Smart -1 (ESA), Hayabusa (JAXA)

6

Hall Thruster Operation Principle

H a ll th ru s te r is a n e le c tric th ru s te r w ith c lo s e d e le c tro n d riftH a ll th ru s te r is a n e le c tric th ru s te r w ith c lo s e d e le c tro n d rift (H a ll c u rre n t)(H a ll c u rre n t)

7

H a l l T h r u s t e r s

8

Hall Thrusters: Some Facts

• Power range – 50 W – 70 kW• Typical specific impulse – 1500 – 2500 sec• Used in space onboard Russian spacecraft since 1972 in

orbit correction propulsion systems• From the 1990-ies research and development of HT

started in US, Europe, Israel etc.• ESA Smart-1 uses Hall thruster as primary propulsion

system

9

Satellites Equipped with Hall Thrusters

Express –A Express –AM

SESAT

GEO telecommunication satellites

10

ESA SMART-1

11

Orbit transfer using low thrust is analyzed for solar electric propulsion systems. The continuous-thrust solutions are extended to the case in which an eclipse shadow arc restricts thrusting in sunlight only, so that thrusting is now intermittent during each orbit. The magnitude of the thrust is assumed to be constant and the direction to be controllable. A numerical solution to the problem of optimal solar electric propulsion orbital transfer is obtained under the general action of Earth oblateness, drag effects and Earth shadowing.

ORBIT TRANSFERORBIT TRANSFER

12

Shadowing Function Shadowing Function

General lighting conditions for a satellite orbiting the Earth.

The umbra, or shadow cone, is the conical shaded region opposite the direction of the Sun in which the entire disk of the Sun is hidden by the disk of the Earth. An eclipse occurs when the spacecraft enters the umbra. In the penumbra, the disk of the Sun is partially, but not wholly, obstructed by the central body disk.

Sun

z*

Sun

z*

Penumbra

y*

Umbra

y*

Penumbra

UmbraEarth

x*Earth

x

Satellite Orbit

*

SunSun

z*

Sun

z*

Penumbra

y*

Umbra

y*

Penumbra

Umbrax*

Satellite Orbit

*Earth

x*Sun

13

Regions of total and partial solar eclipses Regions of total and partial solar eclipses

Γ1Γ3 Γ3

x*

y*

z*

(ρ∗,ϕ∗)Earth Disk

Γ1Γ Γ2

x*

y*

z*

(ρ∗,ϕ∗)Earth Disk

z*

Sun Disk

z*

ΓΓΓΓ2222

ρ*

Earth Disk ΓΓΓΓ1111

ΓΓΓΓ3333

ΓΓΓΓ2222ΓΓΓΓ3333

z*

Sun Disk

z*

ΓΓΓΓ2222

ρ*

Earth Disk

1111

ΓΓΓΓ3333

ΓΓΓΓ2222ΓΓΓΓ3333

ΓΓΓΓ1111

14

Let us now define the function F(ρ*,z*), equal zero in the region Γ1,where a satellite is in total solar eclipse and equal 1 in the region Γ3. In the region Γ2, where a satellite sees a partial solar eclipse, we use a convenient polynomial of degree 3 and make the function F(ρ*,z*) continuous together with its first derivative for arbitrary values of the variable ρ*.

Solar Eclipse Function Solar Eclipse Function FF((ρρ*,z*)*,z*)

ρ∗

1

0

ρ∗

1

0a1(z*) a2(z*)

F(ρ*,z*)

15

With a1 and a2 are defined by

,

Rs is the solar equatorial radius, Re the Earth equatorial radius and s(t) the Earth- Sun distance.

It should be noted that the function F explicitly depends on the time t and the satellite Cartesian coordinates x, y, z.

( ) 2

3

12

1

2

12

1

1

1

23

0

Γ∈

Γ∈

−−−

−−

Γ∈

= *

*

**

*

*,

,

,

ρ

ρ

ρρ

ρ

ρ if

ifaa

a

aa

a

if

F

( ) ees Rz

ts

RRa +−−= *

1 ( ) ees Rz

ts

RRa ++= *

2

Solar Eclipse Function Solar Eclipse Function FF((ρρ*,z*)*,z*)

16

The optimal control problem of orbital transfer with the solar eclipse constraint will be now considered. We will consider more specifically the case of minimum fuel optimal control with the magnitude of the thrust assumed constant and the direction to be controllable.

The system equations of motion are defined by

Where

with gd the disturbance acceleration due to nonsphericity of the Earth, wd the drag acceleration and T the thrust control vector

d3µ ar

v

vr

+−=

=

r

p0ddd mM −

++= Twga

Problem DefinitionProblem Definition

17

In order to include the eclipse effects the following minimum fuel optimization criterion is defined

In this integral expression, F(t) is the shadowing function and is the propellant mass flow rate.

( )[ ]( )∫=

f

0

2p

t

t

dttF

tTmJ

pm

COST FUNCTIONCOST FUNCTION

18

The inclusion of the function F2(t) in the denominator of the penalty function integrand allows to obtain the expression for the optimal switch function considering the solar eclipse constraint. The quality criterion is an approximation of total consumed propellant mass. Propellant mass flow rate is defined by

Where T(t) is the modulus of the thrust vector T, P is the electric power dissipated by the thruster and η is the power system efficiency.

The optimization problem is to find an admissible control T, which minimizes the quality criterion J over a given time interval (t0,tf) and satisfies given initial and final boundary conditions.

ηP

Tmp 2

2

=

19

Let us now define X = [Ω i θ h en em]T as the vector of orbital elements. Ω is the right ascension of the ascending node, i is the inclination, θ is the argument of latitude, h is the angular momentum, en, em are the components of the eccentricity vector on the perpendicular axes lying in the orbital plane. The orbital elements evolution can be described in vector form as

ad = (adr, adθ, adh)T are the non-Keplerian perturbations in the radial, circumferential and normal directions, respectively. The radial direction is along the geocentric radius vector of the spacecraft, the circumferential direction is perpendicular to this radius vector measured positive in the direction of orbital motion and the normal direction is positive along the angular momentum vector of the spacecraft orbit.

daXGX

),(tdt

d =

Lagrange's planetary equationsLagrange's planetary equations

20

The (6x3) matrix G has the form

( )

=

mhmθmr

nhnθnr

hθ

γh

ih

Ωh

00

00

00

00

ψ

GGG

GGG

G

G

G

G

XG ,

21

The equations of motion of a controlled spacecraft in terms of the orbital element set are given by Gauss's form of Lagrange's planetary equations:

dh

θΩa

ih

r

dt

d

sin

sin= dh

θa

h

r

dt

di cos= dθradt

dh =

( )

++

+

+++++= dhdθ

n

ndr

n aiθeθe

θea

θeθe

eθθθa

µ

h

dt

de

mn

m

m tansincos1sin

sincos1cos

cossin

( )

⋅

++

−

+++++−= dh

mn

n

mn

mdr a

iθeθe

θea

θeθe

eθθθa

µ

h

dt

de

tansincos1sin

sincos1sin

sincos dθm

( )( ) dh

mn

mn aiθeθeµ

θh

h

θeθeµ

dt

dθ

tansincos1sinsincos1

3

22

++−++=

22

Instead of time t the independent variable ψ is used, defined by the following differential equation

Where

The use of this independent variable stabilizes the set of ordinary differential equations.

( )( )22

3

sincos1 θeθeµ

h,θ,eh,eK

mn

mnt ++=

( ) ψd,θ,eh,eKdt mnt=

A NEW INDEPENDENT VARIABLEA NEW INDEPENDENT VARIABLE

23

The optimization problem will be solved by the Pontryagin’s maximum principle.

Since the magnitude of the thrust is constant the variable T can be either T0 or zero. The components of the thrust vector Tr, Tθ and Th are constrained by

Using the maximum principle an optimal switching function Sw is defined by,

Where pm is the co-state variable for the propellant mass and p(pr, pθ, ph) the vector of co-state variables, F is the solar eclipse function and Kt defines the independent variable.

PontryaginPontryagin ’’ss Maximum Principle Maximum Principle

20

222 TTTT hθr =++

( )( )pptw mMFpK

pFS

−−=

02

2

1

24

OPTIMAL SOLUTIONOPTIMAL SOLUTION

If

If T = 0

( )0

0

T

TmSw

p≥ 00

Tp

pT =

( )0

0

T

TmSw

p<

25

NUMERICAL RESULTSNUMERICAL RESULTS

Transfer From sun-synchronous 720 km orbitTo sun-synchronous 420 km orbit

RAAN Ω=60.0ºInclination i=98.269ºArgument of perigee ω=0ºEccentricity e=0.00014Mean motion n=14.517 rev/day

RAAN Ω=freeInclination i=97.103ºArgument of perigee ω=free;Eccentricity e=0.000147Mean motion n=15.4888 rev/day

26

APOGEE AND PERIGEE ALTITUDE TIME HISTORY

0 5 10 15 20 25 30 35 40350

400

450

500

550

600

650

700

750

Flight time, day

Alti

tude

at a

poge

e an

d pe

rigee

, km

27

INCLINATION TIME HISTORY INCLINATION TIME HISTORY

0 5 10 15 20 25 30 35 4097

97.2

97.4

97.6

97.8

98

98.2

98.4

Flight time, day

Incl

inat

ion,

deg

28

THRUST COMPONENTS THRUST COMPONENTS

30 30.02 30.04 30.06 30.08 30.1 30.12 30.14 30.16 30.18 30.2-35

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

Flight time, day

Thr

ust c

ompo

nent

s, m

N

Radial

CircumferentialNormal

29

Consumed Propellant Mass Time History Consumed Propellant Mass Time History

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Flight time, day

Con

sum

ed p

rope

llant

mas

s, k

g

30

Orbital IssuesOrbital Issues

Venus Project: Venus Project:

31

The VENµS mission is a joint project of the Israeli Space Agency and theCentre National d’Etudes Spatiales (CNES). It includes two major experiments: in-flight tests of solar electric propulsion in LEO, and high-resolution, multispectral remote sensing of the earth. The first experiment comprises the tests of a propulsion system with Israeli Hall thrusters (IHET) commanded by a completely autonomous onboard control system. It will propel an inter-orbital transfer from the initial LEO 720 km altitude to a final orbit of 420 km altitude and maintain the latter orbit. The mission profile will consist of three stages:

1) Multispectral earth observations from the initial orbit2) Inter-orbital transfer3) Multispectral earth observations from the final orbit

The Asher Space Research Institute is involved in planning the second stage of the mission and in developing the onboard orbital control algorithms.

Venus MissionVenus Mission

32

Dynamics and control of electrically propelled inte r-LEO transfer

The study addresses the two major issues:

• Preflight orbit design;

• Its in-flight implementation

Work Objectives

33

I. Physical PreliminariesI. Physical Preliminaries

34

Dynamic Environment in LEODynamic Environment in LEO

Newtonian gravity ~101 m/s2 Control thrust ~10-4 m/s2

Polar oblateness (J2) ~10-2 m/s2 Thrust exec.error ~10-5 m/s2

Higher zonals, total <10-5 m/s2 Xe mass consumption <10-5 m/s2

Sectorial harmonics ~10-5 m/s2 Aging TBD

Lunar & solar gravity ~10-7 m/s2

Aerodynamic drag <10-6 m/s2

Light pressure ~10-8 m/s2

Natural factorsNatural factors ManMan--caused factorscaused factors

Underlined items can cause secular perturbations.

35

Dynamic Environment in LEO (2)Dynamic Environment in LEO (2)

Polar oblateness dominates among natural perturbations. It causes secular (linear in time) evolution of some orbital elements and leaves other s unchanged except for small oscillations with the orbital period and its multiples.

Higher gravity harmonics in total contribute ~1% to the oscillatory evolution of orbit and to the prece ssion rate.

Air drag causes slow decrease of orbital period and eccentricity.

36

From sun-synchronous 720 km orbit (29 revs per 2 days) To sun-synchronous 420 km orbit (31 revs per 2 days);

Sun-synchronism to be maintained during the transfe r.

Maintenance of synchronism with Earth rotation impossible

Goal: minimum transfer duration

Nominal InterNominal Inter --Orbital TransferOrbital Transfer

37

• The satellite carries two thrusters but not more th an one may operate at the same time

• Thrusters’ axes are fixed in the body frame, thrust direction is controlled by rotations of the satellite as a whole .

•Solar arrays are also fixed in the body frame. Thus , available power correlates with thrust direction.

• Thruster cannot work at the input power below 250 W .

• Thrust, specific impulse, and efficiency depend on the available power.

• Satellite attitudes and therefore thrust directions are subject to constraints.

SEP PeculiaritiesSEP Peculiarities

38

a

-n =π−χ

u

u’

a

-n =π−χ

u

u’

VENuSVENuS SatelliteSatellite

39

Traditionally , design of low thrust trajectories is considered as a variational problem that provides global optimum to thrust control program.

An alternative approach, closed-loop locally optimal guidance algorithm, is preferable

General Approach

40

Optimal control vs. guidance

works in closed loop with navigation

feedback. Thus it is fast and robust with

respect to dynamic model errors;

- easy adaptation to constraints;

the same algorithm can work as the

onboard control algorithm.

It provides locally optimal guidance.

Extra mass and time quite affordable

GuidanceGuidanceOptimal controlOptimal control

works in open loop. The task is formulated

as a 2-point boundary problem. It

- provides globally optimal trajectories

- is slow and sensitive to dynamic model

errors;

- is not well adapted to work with control

or phase state constraints

Computing a transfer as long as ~103 rev

unaffordable onboard.

41

II. Implementation of Guidance algorithmII. Implementation of Guidance algorithm

42

Guidance Algorithm FeaturesGuidance Algorithm Features

1. It deals with orbital elements, not with state vectors.

2. Its output is the thrust direction that provides the fastest evolution of the elements towards their destination values.

2. In order to eliminate the control jitter at approaching the destination orbit, mean elements are used instead of osculating ones

43

Thrust ControllerThrust Controller

When the initial and destination orbits are close to each other, a proper choice of W assures an optimal thrust program, even though the transfer may count many revolutions. Optimal transfer in the space of orbital elements presents as a straight line. This gives the lead to Woptimization. It is done once for all, at preflight preparation phase.e1

e2

e3

E0optim

al transfe

r

Ef

( ) ( );EEWEE T −−= ff2ρ

( )( )

( )( )EE

EEu

T

T

−−

−=∇∇−=

f

f

G

Guu

maxmax 2

2

ρρ

uE

Gdt

d = E – Orbital elements

44

Attitude constraints avoidanceAttitude constraints avoidance

Banned are the attitudes that implya) Telescope axis getting inside the prohibited con e that encircles

the orbital velocity vector;b) Both star trackers having Sun or Earth in their FOV ( with a

margin for straylight)c) Insufficient available power (solar arrays >58 °°°° away from the

sun) .

Normally, the unconstrained optimal attitude does n ot violate more than one constraint in a time, and the violation is weak. Then aconstraint avoidance algorithm computes optimal corrections to the unconstrained attitudes; otherwise it switches the thrust off.

45

Thruster Firing Thruster Firing ConstraintsConstraints

Firing a thruster should not last less than Tmin (=10 minutes).

At real-time operation, the control loop should be capable of control prediction at least Tmin ahead. Though easy to enable in principle, it increases the deman ded computing resources.

An economical algorithm was been developed that use s a simplified dynamic model.

46

66 66.05 66.1 66.15 66.2 66.25 66.3 66.35 66.4400

410

420

430

440

450

460

470

480

490

500

Flight time, day

Alti

tude

at

apog

ee a

nd p

erig

ee,

km

A sample of osculating apogee and perigee heights

External perturbations and control

thrust force oscillatory behavior of

osculating elements. Oscillations

permeate the feedback and cause

parasitic self-exciting oscillations in

the control loop.

Mean elements contain no short-term

oscillations and therefore safe. A

practical way to evaluate them is

averaging the osculating elements over

the period prior to the current instant.

Mean versus osculating elements

47

III. Numerical ResultsIII. Numerical Results

48

Nominal Transfer SummaryNominal Transfer Summary

1.4501.330Xe mass, kg (anode only)

195173Burning time, hours

696615IHET2: On/Off cycles

3.5503.402Xe mass, kg (anode only)

490440Burning time, hours

695616IHET1: On/Off cycles

5.8705.440Xe mass, kg (15% for cathode added)

685612Total burning time, hours

694612Time out of eclipse, hours

4641Transfer time, days

10Constraint Avoidance Strategy

0 = constraints ignored; 1 = constraints avoidance

49

Sample thrust program

Nominal transferNominal transfer

3.00 3.05 3.10 3.15 3.20

-0.20

-0.10

0.00

0.10

0.20

EC

LIP

SE

EC

LIP

SE

EC

LIP

SE

Time, days3.00 3.05 3.10 3.15 3.20

-0.20

-0.10

0.00

0.10

0.20

EC

LIP

SE

EC

LIP

SE

EC

LIP

SE

Con

trol

acc

eler

atio

n, m

m/s

2

50

Sample of power historySample of power historyNominal & Emergency transfers

1.00 1.10 1.20 1.30 1.40Time, days

0

200

400

600In

p utp

ower

,W

nominal case

STR1 in failute

STR2 in failure

IHET1 in failure

51

1. An optimal algorithm for orbit transfer with Sola r Electric Propulsion was developed that explicitly takes into account eclipse constraints

2. A closed loop guidance algorithm was developed th at provides a satisfactory solution to the thrust control and t o its autonomous onboard implementation

3. The guidance algorithm is nearly optimal in terms of transfer duration.

4. The algorithm deals with specially chosen mean e lements. Algorithms for their evaluation from GPS data has b een developed and tested

SUMMARY & CONCLUSIONS

52