Team 5Team 5Moscow State UniversityMoscow State University

Department of Mechanics and Mathematics I.S. Department of Mechanics and Mathematics I.S. Grigoriev, M.P. ZapletinGrigoriev, M.P. Zapletin

zaplet@mail.ru iliagri@mail.ru

3rd Global Trajectory Optimisation Competition

Method of the solutionMethod of the solution

Problem is solved into two stages.Problem is solved into two stages. First stageFirst stage - selection of the scheme of flight: - selection of the scheme of flight:

the selection of the exemplary moments start from thethe selection of the exemplary moments start from theEarth and from the asteroids,Earth and from the asteroids,

the selection of asteroids and order of their visit,the selection of asteroids and order of their visit,the calculation of possible Earth flyby,the calculation of possible Earth flyby,obtaining initial approximation for the second stage.obtaining initial approximation for the second stage.

Selection of the scheme of flight was made on the basis Selection of the scheme of flight was made on the basis of of solution of the problems two-impulse and three-impulse solution of the problems two-impulse and three-impulse optimal flight between the orbits of asteroids, oroptimal flight between the orbits of asteroids, or asteroids asteroids and the Earth, or the Earth and asteroids, and also the and the Earth, or the Earth and asteroids, and also the corresponding of the Lambert problem.corresponding of the Lambert problem.

Second stageSecond stage - solution of the problem of optimal control on - solution of the problem of optimal control on the the basis of Pontryagin’s Maximum Principle for the problems basis of Pontryagin’s Maximum Principle for the problems

with intermediate conditions and parameters.with intermediate conditions and parameters.

3rd Global Trajectory Optimisation Competition

Problem DescriptionProblem DescriptionThe motion of the Earth and asteroids around Sun is governed The motion of the Earth and asteroids around Sun is governed

by these equationsby these equations::

3rd Global Trajectory Optimisation Competition

The boundary conditions:start from the Earth

Arrival to the asteroids i=1,2,3 and the flying awayArrival to the asteroids i=1,2,3 and the flying away

3rd Global Trajectory Optimisation Competition

Arrival to the Еarth:

The total duration of flight is limited:

The calculation Earth flybyThe calculation Earth flyby: :

3rd Global Trajectory Optimisation Competition

A radius of flight is limited:

The boundary-value problem Pontryagin’s The boundary-value problem Pontryagin’s Maximum Principle.Maximum Principle.

3rd Global Trajectory Optimisation Competition

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 → 96Start 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 → 88Start 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.

3rd Global Trajectory Optimisation Competition

88 → 49Start 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 → EarthStart 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

Main pMain publications ublications (in “Cosmic Research”)(in “Cosmic Research”)

K.G. Grigoriev, M.P. Zapletin and D.A. Silaev K.G. Grigoriev, M.P. Zapletin and D.A. Silaev Optimal Insertion of a Spacecraft from the Lunar Surface into a Circular Optimal Insertion of a Spacecraft from the Lunar Surface into a Circular Orbit of a Moon Satellite,1991, vol. 29, no. 5. Orbit of a Moon Satellite,1991, vol. 29, no. 5.

K.G. Grigoriev, E.V. Zapletina and M.P. ZapletinK.G. Grigoriev, E.V. Zapletina and M.P. Zapletin Optimum Spatial Flights of a Spacecraft between the Surface of the Optimum Spatial Flights of a Spacecraft between the Surface of the Moon and Orbit of Its Artificial Satellite, 1993, vol. 31, No. 5. Moon and Orbit of Its Artificial Satellite, 1993, vol. 31, No. 5.

K.G. Grigoriev and I.S. GrigorievK.G. Grigoriev and I.S. Grigoriev Optimal Trajectories of Flights of a Spacecraft with Jet Engine of High Optimal Trajectories of Flights of a Spacecraft with Jet Engine of High Limited Thrust between an Orbits of Artifical Earth Satellites and MoonLimited Thrust between an Orbits of Artifical Earth Satellites and Moon,, 11994, vol. 31, No. 6. 994, vol. 31, No. 6.

K.G. Grigoriev and M.P. Zapletin K.G. Grigoriev and M.P. Zapletin Vertical Start in Optimization Problems of Rocket Dynamics , 1997, vol. Vertical Start in Optimization Problems of Rocket Dynamics , 1997, vol. 35, no. 4. 35, no. 4.

K.G. Grigoriev and I.S. GrigorievK.G. Grigoriev and I.S. Grigoriev Solving Optimization Problems for the Flight Trajectories of a Spacecraft Solving Optimization Problems for the Flight Trajectories of a Spacecraft with a High-Thrust Jet Engine in Pulse Formulation for an Arbitrary with a High-Thrust Jet Engine in Pulse Formulation for an Arbitrary Gravitational Field in a Vacuum, 2002, vol. 40, No. 1. Gravitational Field in a Vacuum, 2002, vol. 40, No. 1.

K.G. Grigoriev and I.S. GrigorievK.G. Grigoriev and I.S. Grigoriev Conditions of the Maximum Principle in the Problem of Optimal Control Conditions of the Maximum Principle in the Problem of Optimal Control over an Aggregate of Dynamic Systems and Their Application to Solution over an Aggregate of Dynamic Systems and Their Application to Solution of the Problems of Optimal Control of Spacecraft Motion, 2003, vol. 41, of the Problems of Optimal Control of Spacecraft Motion, 2003, vol. 41, No. 3. No. 3.

3rd Global Trajectory Optimisation Competition