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launch vehicle inserts the spacecraft (SC) into the low earth orbit

(LEO).

The chemical upper stage (ChUS) inserts the electric propulsion

upper stage (EPS) with payload into the intermediate orbit.

EPS starts from intermediate orbit and delivers the payload intogeostationary orbit (GSO).

launch vehicle inserts the spacecraft (SC) into the low earth orbit

(LEO).

The chemical upper stage (ChUS) inserts the electric propulsion

upper stage (EPS) with payload into the intermediate orbit.

EPS starts from intermediate orbit and delivers the payload intogeostationary orbit (GSO).

Launching stages

Launching stages

The 3rd CSA-IAA Conference on Advanced Space Systems and

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M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion

M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion


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Low thrust trajectory optimization into GEOLow thrust trajectory optimization into GEO

Motion equations are written with the use of equinoctial elements.

Model problem is used to solve the optimization problem of orbital transfer.

Solving method of model problem is based on the use of Pontryagins

maximum principle and averaging method.

The maximum principle is used to solve the main optimization problem also.

Motion equations are written with the use of equinoctial elements.

Model problem is used to solve the optimization problem of orbital transfer.

Solving method of model problem is based on the use of Pontryagins

maximum principle and averaging method.

The maximum principle is used to solve the main optimization problem also.


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M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion


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Boundary conditionsBoundary conditions

It is required to transfer SC initial mass of mo from initial orbit

into final orbit

at time , which is minimized. Minimization of time of flight in

considered statement is equivalent to minimization of fuelconsumption for transfer.

It is required to transfer SC initial mass of mo from initial orbit

into final orbit

at time , which is minimized. Minimization of time of flight in

considered statement is equivalent to minimization of fuelconsumption for transfer.

0 0 0 0 0, , , ,x x y y x x y yA A e e e e i i i i= = = = =

, , , ,k x xk y yk x xk y yk A A e e e e i i i i= = = = =

d


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T

SW

P

Equinoctial elements:Equinoctial elements:

( )+= coseex ( )+= sineey

= cos2

tani

ix = sin2

tani

iy ++= F

=

Motion equations:Motion equations:

T=(P/m)*cos()*cos()T=(P/m)*cos()*cos()

S=(P/m)*sin()*cos()S=(P/m)*sin()*cos()

W=(P/m)* sin()W=(P/m)* sin()

where,


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d

Th 3 d CSA IAA C f Ad d S S d


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Optimal controlOptimal control

Hamiltonian :Hamiltonian :

where, adjoint variables to phase

variables respectively.


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Th 3 d CSA IAA C f Ad d S S d

Th 3rd CSA IAA C f Ad d S S t d


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Optimal control:Optimal control:

By the using of maximum principle, it is possible to get the optimal control

of spacecraft motion (yaw and pitch angles) as follows:

By the using of maximum principle, it is possible to get the optimal control

of spacecraft motion (yaw and pitch angles) as follows:


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Equations for adjoint variablesEquations for adjoint variables


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The 3rd CSA IAA Conference on Advanced Space Systems and


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where ,

acbnon-dimensional jet acceleration.


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Control lawControl law

acceleratingbraking

In work [2] formulated some simplified model problem which offer to receive

analytical results and that results are used as initial guess of a main

problem. Thus it is possible to find the control structure of spacecrafts

motion of orbital transfer. The jet acceleration and yaw angle are consideredas the constants on each revolution of a trajectory.

Structure of control on each revolution of trajectory

On each revolution of a trajectory transversalcomponent of thrust accelerates SC on onearch of a revolution of a trajectory and brakesSC on other arch of a trajectory. Duration andlocation of accelerating arc and braking arc oneach revolution of trajectory are chosen as the

optimized functions of slow time. Binomial component of thrust changes the its

direction in points of an osculating orbit withargument of perigee equal to 90 deg and 270deg concerning a plane of a terminal orbit.

Structure of control on each revolution of trajectory

On each revolution of a trajectory transversalcomponent of thrust accelerates SC on onearch of a revolution of a trajectory and brakesSC on other arch of a trajectory. Duration andlocation of accelerating arc and braking arc oneach revolution of trajectory are chosen as the

optimized functions of slow time. Binomial component of thrust changes the its

direction in points of an osculating orbit withargument of perigee equal to 90 deg and 270deg concerning a plane of a terminal orbit.


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Solving boundary value problemSolving boundary value problem

The boundary value problem of maximum principle for a model

problem is reduced to system of three transcendental equationswith three unknowns. The systems of equations are provided with

satisfaction to reach into a final orbit (semi major axis, eccentricity

and inclination).

The boundary value problem of maximum principle for a model

problem is reduced to system of three transcendental equationswith three unknowns. The systems of equations are provided with

satisfaction to reach into a final orbit (semi major axis, eccentricity

and inclination).

.),,(;5.0),,(

;),,(

0411

0411

0411

iLLLfieLLLfe

A

ALLLfA

ok

ok

k

ok

=

=

=


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fA ,,L 1k L 1o L 4 iff acos L 1k

2L 4

2

f acos L 1o2

L 42


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f( )x sin( )x .

2 x cos ( )x

fe ,,L 1k L 1o L 4 ifd

acos L 1k

acos L 1o

x.sin ( )x

2f( )x

f( )x2

L 42


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fi ,,L 1k L 1o L 4 if.d

acos L 1k

acos L 1o

xsin ( )x

f( )x 2 L 42

L 4


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Relations between ( L1k, L10, L4 ) and adjoint variablesRelations between ( L1k, L10, L4 ) and adjoint variables

where,where,

The 3 CSA IAA Conference on Advanced Space Systems and

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AlgorithmAlgorithm

0. Phase variables (A, i, e) for initial or current point of a trajectory are

known.

0. Phase variables (A, i, e) for initial or current point of a trajectory are

known.

1. Determine the parameters(L1k, L10, L4) and find the values of the

adjoint variables to semi major axis and to an inclination in an initial(current) point and a final point of a trajectory of flight.

1. Determine the parameters(L1k, L10, L4) and find the values of the

adjoint variables to semi major axis and to an inclination in an initial(current) point and a final point of a trajectory of flight.

2. Define the adjoint variables with the use of known parameters.2. Define the adjoint variables with the use of known parameters.

3. Choose the duration of an investigated segment of a trajectory.

For example, one day, or 1 revolution of trajectory around of the Earth.

3. Choose the duration of an investigated segment of a trajectory.

For example, one day, or 1 revolution of trajectory around of the Earth.

4. Integrate the system of the equations of optimum movement of SC on

the chosen interval of time.

4. Integrate the system of the equations of optimum movement of SC on

the chosen interval of time.

5. Parameters of phase variables of an investigated segment of atrajectory are used as initial conditions for a following segment of

trajectory.

5. Parameters of phase variables of an investigated segment of atrajectory are used as initial conditions for a following segment of

trajectory.

Final orbitFinal orbit exitexit


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p y

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Numerical resultsNumerical results

It is considered to insert a satellite into geostationary orbit. Initial intermediateorbit has the following characteristics:

Radius of a perigee of an intermediate orbit - 10000 km, Radius of a apogee of an intermediate orbit - 65000 km,

Inclination of an intermediate orbit - 30 degree.

Argument of a perigee - 0 degrees,

Longitude of the ascending nodes - 0 degrees.Characteristics of a spacecraft in an intermediate orbit:

Initial mass 2000 kg ,

Thrust of electric propulsion engine - 0.4 N,

Specific impulse of electric propulsion engine- 16 km/s.

It is considered to insert a satellite into geostationary orbit. Initial intermediateorbit has the following characteristics:

Radius of a perigee of an intermediate orbit - 10000 km,

Radius of a apogee of an intermediate orbit - 65000 km,

Inclination of an intermediate orbit - 30 degree.

Argument of a perigee - 0 degrees,

Longitude of the ascending nodes - 0 degrees.Characteristics of a spacecraft in an intermediate orbit:

Initial mass 2000 kg ,

Thrust of electric propulsion engine - 0.4 N,

Specific impulse of electric propulsion engine- 16 km/s.

p y

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p y

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Semi major axis (m) as a function of

flight time (day)

p y

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p y

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Radius of perigee and apogee(m) as a

function of flight time (day)

p y

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Components of eccentricity as a

function of flight time (day)






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Eccentricity as a function of flight time

(day)




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A li i


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Inclination (degree) as a function of

flight time (day)





A li ti


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Pitch angle (degree) as a function of

flight time (day)


A li ti


A li ti


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Fig.1 pitch angle (deg) as a function offlight time in 1st revolution of trajectory

Fig.2 pitch angle (deg) as a function of

flight time in 50th

revolution of trajectory

Fig.3 pitch angle (deg) as a function of flight

time in 115th revolution of trajectory




A li ti


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Yaw angle (degree) as a function of flight time

(day)


A li ti


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Fig.1 yaw angle (deg) as a function of

flight time in 1st revolution of trajectoryFig.2 yaw angle (deg) as a function of

argument of perigee in 1st revolution of

trajectory

Fig.3 yaw angle (deg) as a function of

flight time in 115th revolution of trajectoryFig.4 yaw angle (deg) as a function of

argument of perigee in 115th revolution of

trajectory




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3D view of the calculated

transfer trajectory




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The trajectory for geostationary satellite insertion with the use of

chemical upper stage and electric propulsion is designed.

The continuous low thrust trajectory optimization technique of the

multirevolution trajectory from elliptical orbit into noncoplanar

circular orbit is developed.

The results by the use of developed technique, including the

numerical integration of system of equations (phase and adjointvariables) as well as the characteristics of trajectory are presented.

The trajectory for geostationary satellite insertion with the use of

chemical upper stage and electric propulsion is designed.

The continuous low thrust trajectory optimization technique of the

multirevolution trajectory from elliptical orbit into noncoplanar

circular orbit is developed.

The results by the use of developed technique, including the

numerical integration of system of equations (phase and adjointvariables) as well as the characteristics of trajectory are presented.

ConclusionConclusion




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The literaturesThe literatures

1. M. Konstantinov. Optimization of Low Thrust transfer from elliptical

orbit into noncoplanar circular orbit. Proceedings of 2nd

International Symposium on Low-Thrust Trajectories LOTUS-2,

Toulouse, France, 2002.

2. M. Konstantinov. Optimization of low thrust transfer between no

coplanar elliptic orbits. Paper IAF-97-A.6.06, Turin, Italy, October

1997.

3. V. Petukhov. Low-Thrust Trajectory Optimization. Presentation at

the seminar on Space Flight Mechanics, Control, and Information

Science of Space Research Institute (IKI), Moscow, June 14, 2000

(http://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppt)4. M. Konstantinov. Analysis trajectories of the insertion into

geostationary orbit of a SC with the chemical and electric

propulsion with using of Moons swingy.

1. M. Konstantinov. Optimization of Low Thrust transfer from elliptical

orbit into noncoplanar circular orbit. Proceedings of 2nd

International Symposium on Low-Thrust Trajectories LOTUS-2,

Toulouse, France, 2002.2. M. Konstantinov. Optimization of low thrust transfer between no

coplanar elliptic orbits. Paper IAF-97-A.6.06, Turin, Italy, October

1997.

3. V. Petukhov. Low-Thrust Trajectory Optimization. Presentation at

the seminar on Space Flight Mechanics, Control, and Information

Science of Space Research Institute (IKI), Moscow, June 14, 2000

(http://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppt)4. M. Konstantinov. Analysis trajectories of the insertion into

geostationary orbit of a SC with the chemical and electric

propulsion with using of Moons swingy.




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http://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppthttp://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppthttp://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppthttp://arc.iki.rssi.ru/seminar/200006/OLTTE2.ppt


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The endThe end

Thank you!Min Thein

Thank you!Min Thein



presentation for shanghai

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