Институт прикладной математики им. М.В.Келдыша РАН

Институт прикладной математики им. М.В.Келдыша РАН

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

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

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences

Alexey Golikov, Andrey Tuchin

Essential goals


Orbit measurements: interpretation, information processing, ballistic and navigational mission support, etc.o ground supported trajectory measurements (GSTM):• range• range rate o 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

DD

Ganymede Lander: Ganymede Lander: Mission Stages

gravitational maneuvers about Earth & Venus


gravitational maneuvers around Ganymede & Callisto

preliminary elliptical orbit circular polar orbit at the height of 100 km prelanding orbit with low pericenter session on Ganymede’s surface

Artificial satellite of Jupiter

Artificial satellite of Ganymede (ASG)

Launching of the spacecraft (SC) Interplanetary flight Earth → Jupiter

Scheme of the stage ASG


Preliminary orbit

Orbital correctionsGSTM

Orbit period

Inclination

Eccentricity

Descent

Scheme of the stage ASG


1. Transition to preliminary elliptical orbit after braking at approach to Ganymede

2. Series of GSTM for orbit determination3. Orbital corrections of orbit period & inclination to form circular

polar orbit at the height of 100 km4. Series of GSTM within 2 days for orbit determination5. Bound orbital corrections (consisting of 2 corrections of the

orbit period) to precise circular polar orbit6. Circular polar orbit with science experiments7. Orbital maneuver to form a landing orbit8. Series of GSTM on 2-3 adjacent circuits of a landing orbit9. Descent maneuver into given point on the surface of Ganymede

Perturbing forcesPerturbing forces


Essential perturbating factorsEssential perturbating factors


Gravitational field of Ganymede (2×2):

2nd zonal harmonics

2nd sectorial harmonics

Jupiter’s gravity attraction:circular equatorial orbit

Rotation of Ganymede is synchronized with its orbit around Jupiter ,

there are resonance effects

20c

22c

orb rotw w

Preliminary orbitPreliminary 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 e=0.5 it will destroy in 2 hours

4) For eccentricity e<0.3 equatorial elliptical orbits are stable

5) Polar elliptical orbits are unstable for e>0.01

Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5)


Evaluation of the polar orbit Evaluation of the polar orbit (e=0.3)


Transfer to circular polar orbitTransfer to circular polar orbit


Series of maneuvers to change the orbit period & inclination

1) Maneuver optimization by using the Lambert problem with unfixed finite constraints

2) Solution of this problem is achieved by iterative procedure3) Take into consideration an essential condition: the polar

orbit at high altitudes is unstable!4) Supplementary constraint: to form the polar orbit only on

low heights & using “quasiequilibrium points”

Circular polar orbitCircular polar orbit


1) Altitude 100 km2) Series of GSTM within 2 days for orbit determination3) Bound orbital corrections (consisting 2 corrections of

the orbit period) to precise circular polar orbit4) Science experiments (with orbit keeping corrections)5) It needs to take into account the orbit evolution

(perturbations)6) Orbital maneuvers to form a prelanding orbit with low

pericenter

90 , 0i e

Circular polar orbitCircular polar orbit


Long-periodic perturbations of the orbit: 90 , 0i e

22

20

2 2

1cos 2

16cos 2 cos 215 21 cos 2

32

G

e ee c R a

W We e

22

22 2

22

2

3 2 3 3cos 2 cos 22 161

cos 2 cos 21532 1

Gei c R p W We

W We

e

0 ,rotw n where 220

3 cos ,2 Gi c R p 2 2

203 1 5cos4 Gc R p i

Evaluation of polar circular orbitEvaluation of polar circular orbit


Prelanding orbitPrelanding orbit


1) Altitude of the pericenter 15 km2) Altitude of the apocenter 100 km3) Eccentricity 0.01584) Series of GSTM on 2-3 adjacent circuits of a landing

orbit to precise orbital parameters5) 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

Evaluation of prelanding orbitEvaluation of prelanding orbit


Descent SessionDescent Session


• 2 variants depending on the start time of descent:1) 24 hours => 16 hours of measurements GSTM2) 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

Mass before descent maneuver 900 kg

Propulsion system 215 kg

Total burn 4200 N

Specific thrust 319 s

Dry mass 385 kg

Ganymede Lander moduleGanymede Lander module


Example of solutionExample 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

Direction of the ThrustDirection of the Thrust


Altitude vs. DistanceAltitude vs. Distance


Velocity vs. TimeVelocity vs. Time


Radial velocityRadial velocity


Transversal velocityTransversal velocity


Thank you!Thank you!

Alexei R. GolikovAlexei R. [email protected]

Andrey G. TuchinAndrey G. [email protected]

Keldysh Institute of Applied Mathematics,Russian Academy of Sciences


Институт прикладной математики им. М.В.Келдыша РАН

Documents

scientific goal

orbit measurements

kmprelanding orbit

thrusterganymede lander

earth venus ganymede

stage asgganymede lander

ganymedeseries of gstm

scheme of maneuvers