![Page 1: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/1.jpg)
Институт прикладной математики им. М.В.Келдыша РАН
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
![Page 2: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/2.jpg)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences
Alexey Golikov, Andrey Tuchin
![Page 3: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/3.jpg)
Essential goals
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 4: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/4.jpg)
Ganymede Lander: Ganymede Lander: Mission Stages
gravitational maneuvers about Earth & Venus
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 5: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/5.jpg)
Scheme of the stage ASG
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
Preliminary orbit
Orbital correctionsGSTM
Orbit period
Inclination
Eccentricity
Descent
![Page 6: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/6.jpg)
Scheme of the stage ASG
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 7: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/7.jpg)
Perturbing forcesPerturbing forces
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 8: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/8.jpg)
Essential perturbating factorsEssential perturbating factors
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 9: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/9.jpg)
Preliminary orbitPreliminary orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 10: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/10.jpg)
Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 11: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/11.jpg)
Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 12: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/12.jpg)
Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.5)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 13: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/13.jpg)
Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.3)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 14: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/14.jpg)
Evaluation of the polar orbit Evaluation of the polar orbit (e=0.3)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 15: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/15.jpg)
Evaluation of preliminary orbit Evaluation of preliminary orbit (e=0.1)
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 16: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/16.jpg)
Transfer to circular polar orbitTransfer to circular polar orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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”
![Page 17: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/17.jpg)
Circular polar orbitCircular polar orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 18: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/18.jpg)
Circular polar orbitCircular polar orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 19: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/19.jpg)
Evaluation of polar circular orbitEvaluation of polar circular orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 20: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/20.jpg)
Evaluation of polar circular orbitEvaluation of polar circular orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 21: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/21.jpg)
Prelanding orbitPrelanding orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
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
![Page 22: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/22.jpg)
Evaluation of prelanding orbitEvaluation of prelanding orbit
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 23: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/23.jpg)
Descent SessionDescent Session
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
• 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
![Page 24: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/24.jpg)
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
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 25: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/25.jpg)
Example of solutionExample of solution
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
• 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
![Page 26: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/26.jpg)
Direction of the ThrustDirection of the Thrust
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 27: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/27.jpg)
Altitude vs. DistanceAltitude vs. Distance
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 28: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/28.jpg)
Velocity vs. TimeVelocity vs. Time
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 29: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/29.jpg)
Radial velocityRadial velocity
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 30: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/30.jpg)
Transversal velocityTransversal velocity
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013
![Page 31: Институт прикладной математики им. М.В.Келдыша РАН](https://reader035.vdocuments.site/reader035/viewer/2022062315/56815e81550346895dcd133f/html5/thumbnails/31.jpg)
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
“Ganymede Lander: scientific goal and experiments”, 5-7 March 2013