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Team 5

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

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Team 5

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  1. 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

  2. 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 Pontryagin’s Maximum Principle for the problems with intermediate conditions and parameters.

  3. 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

  4. 3rd Global Trajectory Optimisation Competition Arrival to the asteroids i=1,2,3 and the flying away Arrival to the Еarth: The total duration of the flight is limited:

  5. 3rd Global Trajectory Optimisation Competition The calculation Earth flyby: A radius of the flight is limited:

  6. 3rd Global Trajectory Optimisation Competition The boundary-value problem of Pontryagin’s Maximum Principle.

  7. 3rd Global Trajectory Optimisation Competition

  8. 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.

  9. 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.

  10. 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

  11. 3rd Global Trajectory Optimisation Competition Main publications (in “Cosmic Research”) • 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. ZapletinOptimum 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. GrigorievOptimal 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. GrigorievSolving 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. GrigorievConditions 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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