Moscow State University

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Team 5
3rd Global Trajectory Optimisation Competition

• Moscow State University
• Department of Mechanics and Mathematics
  I.Grigoriev, J.Plotnikova, M.Zapletin, E.Zapletina

zaplet_at_mail.ru iliagri_at_mail.ru
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3rd Global Trajectory Optimisation Competition

• Method of the solution
• Problem is solved into two stages.
• First stage 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 the solution of the optimal
  control problem
• on the basis of Pontryagins Maximum Principle
  for the problems with intermediate conditions
  and parameters.

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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
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Arrival to the asteroids i1,2,3 and the flying
away
3rd Global Trajectory Optimisation Competition
Arrival to the ?arth
The total duration of the flight is limited
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The calculation Earth flyby
3rd Global Trajectory Optimisation Competition
A radius of the flight is limited
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The boundary-value problem of Pontryagins
Maximum Principle.
3rd Global Trajectory Optimisation Competition
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3rd Global Trajectory Optimisation Competition
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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.
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3rd Global Trajectory Optimisation Competition
Earth ? 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 - tf1 222.553 Day.
96 ? Earth flyby ? 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. ts2 - tf2 165.170
Day.
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3rd Global Trajectory Optimisation Competition
88 ? 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 -
tf3 1616.428 Day
49 ? 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
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Main publications (in
Cosmic Research)
3rd Global Trajectory Optimisation Competition

• 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.