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Институт прикладной математики им. М.В.Келдыша РАН. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences. Mathematical Model of the Spacecraft Landing on Ganymede’s Surface. Alexey Golikov, Andrey Tuchin. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences.

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Институт прикладной математики им. М.В.Келдыша РАН

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

alexey golikov andrey tuchin

Mathematical Model

of the Spacecraft Landing

on Ganymede’s Surface

Alexey Golikov, Andrey Tuchin

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

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

essential goals
Essential goals
  • 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 the thruster

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

ganymede lander mission stages
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 height of 100 km
  • prelanding orbit with low pericenter
  • session on Ganymede’s surface

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

scheme of the stage asg
Scheme of the stage ASG

Preliminary orbit

Orbital corrections

GSTM

Orbit period

Inclination

Eccentricity

_________

_____

Descent

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

scheme of the stage asg1
Scheme of the stage ASG

Transition to preliminary elliptical orbit after braking at approach to Ganymede

Series of GSTM for orbit determination

Orbital corrections of orbit period & inclination to form circular polar orbit at the height of 100 km

Series of GSTM within 2 days for orbit determination

Bound orbital corrections (consisting of 2 corrections of the orbit period) to precise circular polar orbit

Circular polar orbit with science experiments

Orbital maneuver to form a landing orbit

Series of GSTM on 2-3 adjacent circuits of a landing orbit

Descent maneuver into given point on the surface of Ganymede

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

perturbing forces
Perturbing forces

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

essential perturbating factors
Essential perturbating factors
  • Gravitational field of Ganymede (2×2):

2nd zonal harmonics

2ndsectorial harmonics

  • Jupiter’s 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

preliminary orbit
Preliminary orbit

Near equatorial and high eccentric orbit

Take into account the orbit evolution (perturbations)

Preliminary orbit with high eccentricity is very unstable: for e=0.5 it will destroy in 2 hours

For eccentricity e<0.3 equatorial elliptical orbits are stable

Polar elliptical orbits are unstable for e>0.01

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

evaluation of preliminary orbit e 0 5
Evaluation of preliminary orbit (e=0.5)

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

evaluation of preliminary orbit e 0 51
Evaluation of preliminary orbit (e=0.5)

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

evaluation of preliminary orbit e 0 52
Evaluation of preliminary orbit (e=0.5)

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

evaluation of preliminary orbit e 0 3
Evaluation of preliminary orbit (e=0.3)

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

evaluation of the polar orbit e 0 3
Evaluation of the polar orbit (e=0.3)

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

evaluation of preliminary orbit e 0 1
Evaluation of preliminary orbit (e=0.1)

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

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

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

circular polar orbit
Circular polar orbit

Altitude 100 km

Series of GSTM within 2 days for orbit determination

Bound orbital corrections (consisting 2 corrections of the orbit period) to precise circular polar orbit

Science experiments (with orbit keeping corrections)

It needs to take into account the orbit evolution (perturbations)

Orbital maneuvers to form a prelanding orbit with low pericenter

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

circular polar orbit1
Circular polar orbit

Long-periodic perturbations of the orbit:

where

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

evaluation of polar circular orbit
Evaluation of polar circular orbit

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

evaluation of polar circular orbit1
Evaluation of polar circular orbit

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

prelanding orbit
Prelanding orbit

Altitude of the pericenter 15 km

Altitude of the apocenter 100 km

Eccentricity 0.0158

Series of GSTM on 2-3 adjacent circuits of a landing orbit to precise orbital parameters

Limit errors of GSTM are non greater than 0.2 mm/s and 20 m

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

evaluation of prelanding orbit
Evaluation of prelanding orbit

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

descent session
Descent Session
  • 2 variants depending on the start time of descent:
  • 24 hours => 16 hours of measurements GSTM
  • 12 hours => 6 hours of measurements GSTM
  • Nominal program of the thrust direction corresponds to the solution of the problem optimization
  • Using Pontryagin’s 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

ganymede lander module
Ganymede Lander module
  • Mass before descent maneuver 900 kg
  • 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

example of solution
Example of solution
  • Solution by Pontryagin’s 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

direction of the thrust
Direction of the Thrust

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

altitude vs distance
Altitude vs. Distance

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

velocity vs time
Velocity vs. Time

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

radial velocity
Radial velocity

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

transversal velocity
Transversal velocity

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

thank you
Thank you!

Alexei R. Golikov

[email protected]

Andrey G. Tuchin

[email protected]

Keldysh Institute of Applied Mathematics,

Russian Academy of Sciences

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

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