sergey trofimov keldysh institute of applied mathematics, ras

Method of Virtual Trajectories for the Preliminary Design of Multiple Gravity-Assist Interplanetary

Trajectories

Sergey TrofimovKeldysh Institute of Applied Mathematics, RAS

Moscow Institute of Physics and Technology

Michael OvchinnikovKeldysh Institute of Applied Mathematics, RAS

Maksim ShirobokovKeldysh Institute of Applied Mathematics, RAS

Moscow Institute of Physics and Technology

64th International Astronautical Congress (IAC) 23-27 September 2013, Beijing, China

Contents

• Motivation for inventing a method

• Method of virtual trajectories (MVT)

• Benefits and flaws of the MVT

• Test case: Flight to Jupiter

• Conclusions

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Mission feasibility studyWhen studying the mission feasibility, a designer wants:• To quickly estimate the best V, the transfer time and

launch windows for a number of planetary sequences• To have an option of varying some mission constraints

and imposing new ones (ideally without repeating the whole optimization procedure)

• To do all of this without involving skilled specialists in astrodynamics

These demands are difficult to meet in case of multiple gravity-assist (MGA) trajectory design

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Method of virtual trajectories• Based on the fact that the orbits of planets are

changing very slowly

• For a given planetary sequence, a database of all “geometrically feasible” trajectories can be constructed once and for all (“for all” means at least for several decades)

• The second, fast computing step: to screen and refine such a database of virtual trajectories

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Classes of trajectories consideredBasic class of trajectories:

• Coast heliocentric conic arcs

• Powered gravity assists (single impulse at the pericenter)

Method of VT was also adapted to the trajectories with

• non-powered gravity assists

• deep space maneuvers (DSMs)

At most one DSM is allowed on each heliocentric arc

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Some basic concepts and assumptions1) The orbits of planets:• assumed to be closed curves fixed in space• are discretized (i.e., represented as a 1D mesh)

2) Virtual trajectory (VT):• consists of heliocentric conic arcs• sequentially connecting the mesh points on the orbits of planets

included in the planetary sequence chosen3) A virtual trajectory is referred to as near-feasible if a spacecraft

moving along it would fly by the mesh node on the planet’s orbit approximately (within some time tolerance) at the same time with the planet itself

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Discretization of planetary orbitsand beams of virtual trajectories

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2

1

1 2

1 1 cos2cos cos cospar

vv r r


Patching of incoming and outgoing planetocentric hyperbolic arcs

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Powered GA maneuvers Unpowered GA maneuvers


Screening of a VT database and refinement of near-feasible trajectories

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Pruning infeasible trajectories Refinement of near-feasible ones


Comparison of computational costs

Number of gravity

assists

CPU time for VT database screening and refinement,

min*

CPU time for standard Lambert-based approach,

min*

1 0.5-2 2-32 3-6 10-153 8-15 60-804 20-40 >200

*All values of computational time are relative to a PC with 2.13 GHz CPU and 2Gb RAM

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Benefits and flaws of the VT method+ One and the same set of databases can be used

many times for the design of various missions+ Easy handles with imposing different additional

constraints, without extra computational cost− Sensitive to step sizes during the discretization

of planets’ orbits− Requires considerable hard disk space for saving

all the VT databases (from 10 MB up to 1 GB for a long planetary sequence with 5 GAs)

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Sample problem: Transfer to JupiterObjective function:

Constraints:

No conjunctions during performing GAs or DSMs

To check some standard planetary sequences: EVJ, EVEJ, EEVJ, EVEEJ

minV

2020,2025launchT

3 km/sV

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EVEEJ with powered GA maneuvers

194 m/s6.02 yrs

11 / 03 / 2020flight

launch

VT

t

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EVEEJ with DSMs and unpowered GAs

88 m/s6.03 yrs

13 / 03 / 2020flight

launch

VT

t

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This trajectory is similar to the baseline trajectory of Jupiter Ganymede Orbiter (JGO) mission


Comparison of trajectories obtained usingthe MVT with DSMs and in the JGO mission

JGO trajectory MVT with DSMsLaunch 11/03/2020 13/03/2020

Venus flyby 01/07/2020 30/06/2020First Earth flyby 27/04/2021 27/04/2021

Second Earth flyby 28/07/2023 28/07/2023Jupiter approach 04/02/2026 25/03/2026

in EV 0 0.01 in VE 0 0.07 in EE 39 88 in EJ 0 0.4

Escape velocity, km/s 3.39 3.41Approach velocity , km/s 5.50 5.58

Duration, year 5.9 6.0

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ConclusionsBased on a number of beforehand computed databases of virtual trajectories, a mission designer can:

• quickly estimate the possible mission timeline options (planetary sequence, launch date, transfer time)

• pick and choose the planetary sequence which is best suited to various constraints and scientific requirements

• change his mind and impose new constraints without a serious increase in time of mission feasibility analysis

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Acknowledgments

• Russian Academy of Sciences (RAS), Presidium

Program “Fundamental Issues in Investigation

and Exploration of Solar System”, Subprogram

“Mission Scenarios and Trajectory Design”

• Russian Foundation for Basic Research (RFBR),

Grant No. 13-01-00665

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sergey trofimov keldysh institute of applied mathematics, ras

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