Presentation for Shanghai
Transcript of Presentation for Shanghai
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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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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.
The 3rd CSA-IAA Conference on Advanced Space Systems and
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The 3rd CSA-IAA Conference on Advanced Space Systems and
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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The 3rd CSA-IAA Conference on Advanced Space Systems and
Applications
The 3rd CSA-IAA Conference on Advanced Space Systems and
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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,
The 3rd CSA-IAA Conference on Advanced Space Systems and
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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
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.
The 3rd CSA-IAA Conference on Advanced Space Systems and
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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
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:
The 3rd CSA-IAA Conference on Advanced Space Systems and
Applications
The 3rd CSA-IAA Conference on Advanced Space Systems and
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
Th 3rd CSA IAA C f Ad d S S t d
Th 3rd CSA IAA C f Ad d S S t d
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Equations for adjoint variablesEquations for adjoint variables
The 3rd CSA-IAA Conference on Advanced Space Systems and
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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
Th 3rd CSA IAA C f Ad d S S t d
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where ,
acbnon-dimensional jet acceleration.
The 3rd CSA-IAA Conference on Advanced Space Systems and
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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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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.
The 3rd CSA-IAA Conference on Advanced Space Systems and
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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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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
=
=
=
The 3rd CSA-IAA Conference on Advanced Space Systems and
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The 3 d 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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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,
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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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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
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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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
Applications
p y
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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Semi major axis (m) as a function of
flight time (day)
p y
Applications
p y
Applications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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Radius of perigee and apogee(m) as a
function of flight time (day)
p y
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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Components of eccentricity as a
function of flight time (day)
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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Eccentricity as a function of flight time
(day)
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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Inclination (degree) as a function of
flight time (day)
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
Pitch angle (degree) as a function of
flight time (day)
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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
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
Yaw angle (degree) as a function of flight time
(day)
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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
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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3D view of the calculated
transfer trajectory
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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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
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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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.
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion
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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
ApplicationsApplications
M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion M. Konstantinov , M. Thein: Trajectory design for geostationary satellite insertion