1. Team 5 Moscow State University Department of Mechanics and Mathematics I.S. Grigoriev, M.P. Zapletin zaplet@mail.ruzaplet@mail.ru iliagri@mail.ru 3rd Global Trajectory Optimisation Competition

2. Method of the solution Problem is solved into two stages. First stage - selection of the scheme of flight: First stage - selection of the scheme of flight: the selection of the exemplary moments start from the Earth and from the asteroids, the selection of asteroids and order of their visit, the calculation of 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, orasteroids and the Earth, or the Earth and asteroids, and also the corresponding of the Lambert problem. Second stage - solution of the problem of optimal control on the basis of Pontryagin’s Maximum Principle for the problems with intermediate conditions and parameters. Second stage - solution of the problem of optimal control on the basis of Pontryagin’s Maximum Principle for the problems with intermediate conditions and parameters. 3rd Global Trajectory Optimisation Competition

3. Problem Description The motion of the Earth and asteroids around Sun is governed by these equations : 3rd Global Trajectory Optimisation Competition The boundary conditions: start from the Earth

4. Arrival 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:

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

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

8. 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. 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, 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, 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 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 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 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. 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. 3rd Global Trajectory Optimisation Competition