Method of Virtual Trajectories for the Preliminary
Design of Multiple Gravity-Assist Interplanetary
Trajectories
Sergey Trofimov
Keldysh Institute of Applied Mathematics, RAS
Moscow Institute of Physics and Technology
Michael Ovchinnikov
Keldysh Institute of Applied Mathematics, RAS
Maksim Shirobokov
Keldysh Institute of Applied Mathematics, RAS
Moscow Institute of Physics and Technology
Contents
• Motivation for inventing a method
• Method of virtual trajectories (MVT)
• Benefits and flaws of the MVT
• Test case: Flight to Jupiter
• Conclusions
64th International Astronautical Congress (IAC)
23-27 September 2013, Beijing, China
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Mission feasibility study
When 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
64th International Astronautical Congress (IAC)
23-27 September 2013, Beijing, China
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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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23-27 September 2013, Beijing, China
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Classes of trajectories considered
Basic 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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23-27 September 2013, Beijing, China
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Some basic concepts and assumptions
1) 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 chosen
3) 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 orbits
and beams of virtual trajectories
2
 v1 
1
1  cos 



 v par  2cos   r1 r2  cos   cos    
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Patching of incoming and outgoing
planetocentric hyperbolic arcs
Powered GA maneuvers
Unpowered GA maneuvers
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Screening of a VT database and
refinement of near-feasible trajectories
Pruning infeasible trajectories
Refinement of near-feasible ones
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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
2-3
2
3-6
10-15
3
8-15
60-80
4
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 Jupiter
Objective function: V  min
Constraints:
Tlaunch  2020, 2025
V  3 km/s
No conjunctions during performing GAs or DSMs
To check some standard planetary sequences: EVJ,
EVEJ, EEVJ, EVEEJ
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EVEEJ with powered GA maneuvers
V  194 m/s
Tflight  6.02 yrs
tlaunch  11 / 03 / 2020
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EVEEJ with DSMs and unpowered GAs
V  88 m/s
Tflight  6.03 yrs
tlaunch  13 / 03 / 2020
This trajectory is similar to the
baseline trajectory of Jupiter
Ganymede Orbiter (JGO) mission
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Comparison of trajectories obtained using
the MVT with DSMs and in the JGO mission
JGO trajectory
MVT with DSMs
Launch
11/03/2020
13/03/2020
Venus flyby
01/07/2020
30/06/2020
First Earth flyby
27/04/2021
27/04/2021
Second Earth flyby
28/07/2023
28/07/2023
Jupiter approach
04/02/2026
25/03/2026
𝛥𝑉 in EV
0
0.01
𝛥𝑉 in VE
0
0.07
𝛥𝑉 in EE
39
88
𝛥𝑉 in EJ
0
0.4
Escape velocity, km/s
3.39
3.41
Approach velocity , km/s
5.50
5.58
Duration, year
5.9
6.0
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Conclusions
Based 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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