Formation Flying

1
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Keldysh Institute of Applied Mathematics, Russian
Academy of Sciences
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Mathematical Model of the Spacecraft Landing on
Ganymedes Surface

• Alexey Golikov, Andrey Tuchin

Keldysh Institute of Applied Mathematics, Russian
Academy of Sciences
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
3
Essential objectives

• Orbit measurements interpretation, information
  processing, ballistic and navigational mission
  support, etc.
• ground supported trajectory measurements (GSTM)
• range
• range rate
• measurements by the strup down
• Orbit determination determination of all orbital
  parameters taken into account essential orbit
  perturbations
• Maneuver optimization planning the scheme of
  maneuvers, error estimation of maneuver
  realization
• Landing on the surface of Ganimede optimal
  scheme of descent session by using of thrusters

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Ganymede Lander Mission Stages

• Launching of the spacecraft (SC)
• Interplanetary flight Earth ? Jupiter

• gravitational maneuvers about Earth Venus

• Artificial satellite of Jupiter

• gravitational maneuvers around Ganymede
  Callisto

• Artificial satellite of Ganymede (ASG)

• preliminary elliptical orbit
• circular polar orbit at the altitude of 100 km
• prelanding orbit with low pericenter
• session on Ganymedes surface

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Scheme of the stage ASG
Preliminary orbit
Orbital corrections
GSTM
Orbit period
Inclination
Eccentricity
Prelanding orbit
Descent
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Scheme of the stage ASG

1. Transition to preliminary elliptical orbit after
   braking at approach to Ganymede
2. Series of GSTM for orbit determination
3. Orbital corrections of orbit period inclination
   to form circular polar orbit at the altitude of
   100 km
4. Series of GSTM within 2 days for orbit
   determination
5. Bound orbital corrections (consisting of 2
   corrections of the orbit period) to precise
   circular polar orbit
6. Circular polar orbit with science experiments
7. Orbital maneuver to form prelanding orbit
8. Series of GSTM on 2-3 adjacent circuits of
   prelanding orbit
9. Descent maneuver into given point on the surface
   of Ganymede

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Perturbing forces
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Essential perturbating factors

• Gravitational field of Ganymede (22)
• 2nd zonal harmonics
• 2nd sectorial harmonics
• Jupiters gravity attraction
• circular equatorial orbit
• Rotation of Ganymede is synchronized with its
  orbit around Jupiter ,
  
• there are resonance effects

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Preliminary orbit

1. Near equatorial and high eccentric orbit
2. Take into account the orbit evolution
   (perturbations)
3. Preliminary orbit with high eccentricity is very
   unstable for e0.5 it will destroy in 2 hours
4. For eccentricity elt0.3 equatorial elliptical
   orbits are stable
5. Polar elliptical orbits are unstable for egt0.01

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of preliminary orbit (e0.5)
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of preliminary orbit (e0.5)
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of preliminary orbit (e0.5)
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of preliminary orbit (e0.3)
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of the polar orbit (e0.3)
Ganymede Lander scientific goal and
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Evaluation of preliminary orbit (e0.1)
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experiments, 5-7 March 2013
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Transfer to circular polar orbit

• Series of maneuvers to change the orbit period
  inclination
• Maneuver optimization by using Lamberts problem
  with unfixed finite constraints
• Solution of this problem is achieved by iterative
  procedure
• Take into consideration an essential condition
  polar orbit at high altitudes is unstable!
• Supplementary constraint to form polar orbit
  only on low heights using quasiequilibrium
  points

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Circular polar orbit

1. Altitude 100 km
2. Series of GSTM within 2 days for orbit
   determination
3. Bound orbital corrections (consisting 2
   corrections of the orbit period) to precise
   circular polar orbit
4. Science experiments (with orbit keeping
   corrections)
5. It needs to take into account the orbit evolution
   (perturbations)
6. Orbital maneuvers to form prelanding orbit with
   low pericenter

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Circular polar orbit
Long-periodic perturbations of the orbit
where
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of polar circular orbit
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of polar circular orbit
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Prelanding orbit

1. Altitude of the pericenter 15 km
2. Altitude of the apocenter 100 km
3. Eccentricity 0.0158
4. Series of GSTM on 2-3 adjacent circuits of
   prelanding orbit to precise orbital parameters
5. Limit errors of GSTM are non greater than 0.2
   mm/s and 20 m
6. Preliminary estimated errors of orbit prediction
   at the start of descent are non greater 2.5 m/s
   and 5 km

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Evaluation of prelanding orbit
Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Descent Session

• 2 variants depending on the start time of
  descent
• 24 hours gt 16 hours of measurements GSTM
• 12 hours gt 6 hours of measurements GSTM
• Nominal program of the thrust direction
  corresponds to the solution of the problem
  optimization
• Using Pontryagins principle of maximum
• Constraints depend on the problem definition
• Navigation is provided by the strup down

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Ganymede Lander module

• Mass before descent maneuver 900 kg
• Mass of propulsion system 215 kg
• Total burn 4200 N
• Specific thrust 319 s
• Dry mass 385 kg

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Example of solution

• Solution by Pontryagins principle of maximum
• First stage of the descent session from 15 km to
  2 km
• Results of solution
• vertical velocity 10 m/s forward to center of
  Ganymede
• descent duration 320 sec
• fuel expenses 422 kg
• angle distance of descent 7.4 deg

Ganymede Lander scientific goal and
experiments, 5-7 March 2013
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Direction of the Thrust
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experiments, 5-7 March 2013
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Altitude vs. Distance
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Velocity vs. Time
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experiments, 5-7 March 2013
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Radial velocity
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Transversal velocity
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Contacts

• Alexei R. Golikov
• golikov_at_keldysh.ru
• Andrey G. Tuchin
• tag_at_keldysh.ru
• Keldysh Institute of Applied Mathematics,
• Russian Academy of Sciences

Ganymede Lander scientific goal and
experiments, 5-7 March 2013