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# Team 5 presentation

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```3rd Global Trajectory Optimisation Competition
Team 5
Moscow State University
Department of Mechanics and Mathematics
I.Grigoriev, J.Plotnikova, M.Zapletin, E.Zapletina
zaplet@mail.ru
iliagri@mail.ru
3rd Global Trajectory Optimisation Competition
Method of the solution
Problem is solved into two stages.
First stage вЂ“ the selection of the scheme of the flight:
the selection of the exemplary moments of the start from
the Earth and from the asteroids,
the selection of asteroids and the order of their visits,
taking into account of the possible Earth flyby,
obtaining initial approximation for the second stage.
Selection of the scheme of flight was made on the basis of
solution of the problems two-impulse and three-impulse
optimal flight between the orbits of asteroids, or asteroids
and the Earth, or the Earth and asteroids, and also the
corresponding the Lambert problem.
Second stage вЂ“ the solution of the optimal control problem
on the basis of PontryaginвЂ™s Maximum Principle for the
problems with intermediate conditions and parameters.
3rd Global Trajectory Optimisation Competition
Problem Description
The motion of the Earth and asteroids around the Sun is
governed by these equations:
The boundary conditions:
start from the Earth
3rd Global Trajectory Optimisation Competition
Arrival to the asteroids i=1,2,3 and the flying away
Arrival to the Р•arth:
The total duration of the flight is limited:
3rd Global Trajectory Optimisation Competition
The calculation Earth flyby:
A radius of the flight is limited:
3rd Global Trajectory Optimisation Competition
The boundary-value problem of PontryaginвЂ™s
Maximum Principle.
3rd Global Trajectory Optimisation Competition
3rd Global Trajectory Optimisation Competition
Optimality conditions Earth flyby:
The boundary-value problem was solved by a shooting method based
on a modified Newton method and the method of the continuation on
parameters.
3rd Global Trajectory Optimisation Competition
Earth в†’ 96
Start in Earth: 58478.103 MJD.
Passive arc 46.574 Day,
thrust arc 38.513 Day,
passive arc 14.880 Day,
thrust arc 212.446 Day.
Finish in Asteroid 96: 58790.517
MJD.
Mass SC: 1889.448 kg.
ts1 в€’ tf1 = 222.553 Day.
96 в†’ Earth flyby в†’ 88
Start in Asteroid 96: 59013.069 MJD.
Thrust arc 98.649 Day,
passive arc 40.778 Day.
Earth flyby: 59152.497 MJD.
Rp = 6871.000 km.
Passive arc 117.524 Day,
thrust arc 51.617 Day,
passive arc 130.726 Day,
thrust arc 176.911 Day.
Finish in Asteroid 88: 59629.273 MJD.
Mass SC: 1745.321 kg.
3rd Global Trajectory Optimisation Competition
88 в†’ 49
Start in Asteroid 88: 59794.443
MJD.
Thrust arc 63.146 Day,
passive arc 82.524 Day,
thrust arc 58.564 Day,
passive arc 141.733 Day,
thrust arc 28.810 Day.
Finish in Asteroid 49: 60169.221
MJD.
Mass SC: 1679.014 kg.
ts3 в€’ tf3 = 1616.428 Day
49 в†’ Earth
Start in Asteroid 49: 61785.650
MJD.
Thrust arc 26.478 Day,
passive arc 108.492 Day,
thrust arc 77.253 Day.
Mass SC: 1633.319 kg.
Total flight time 3519.769 Day.
Objective function J = 0.82570369
3rd Global Trajectory Optimisation Competition
Main publications
(in вЂњCosmic ResearchвЂќ)
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K.G. Grigoriev, M.P. Zapletin and D.A. Silaev
Optimal Insertion of a Spacecraft from the Lunar Surface into a Circular
Orbit of a Moon Satellite,1991, vol. 29, no. 5.
K.G. Grigoriev, E.V. Zapletina and M.P. Zapletin
Optimum Spatial Flights of a Spacecraft between the Surface of the
Moon and Orbit of Its Artificial Satellite, 1993, vol. 31, No. 5.
K.G. Grigoriev and I.S. Grigoriev
Optimal Trajectories of Flights of a Spacecraft with Jet Engine of High
Limited Thrust between an Orbits of Artifical Earth Satellites and Moon,
1994, vol. 31, No. 6.
K.G. Grigoriev and M.P. Zapletin
Vertical Start in Optimization Problems of Rocket Dynamics , 1997, vol.
35, no. 4.
K.G. Grigoriev and I.S. Grigoriev
Solving Optimization Problems for the Flight Trajectories of a Spacecraft
with a High-Thrust Jet Engine in Pulse Formulation for an Arbitrary
Gravitational Field in a Vacuum, 2002, vol. 40, No. 1.
K.G. Grigoriev and I.S. Grigoriev
Conditions of the Maximum Principle in the Problem of Optimal Control
over an Aggregate of Dynamic Systems and Their Application to Solution
of the Problems of Optimal Control of Spacecraft Motion, 2003, vol. 41,
No. 3.
```
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