tube dynamics and low energy trajectory from the …sdross/talks/onozaki-yoshimura-ross...tube...

Tube Dynamics and Low Energy Trajectory from the Earth to the Moon

in the Coupled Three-Body System

○Kaori Onozaki*, Hiroaki Yoshimura* and Shane D. Ross***Waseda University

Applied Mechanics and Aerospace Engineering

**Virginia Polytechnic Institute and State University Department of Engineering Science and Mechanics

16th International Conference on Astrodynamics Tools and Techniques

March 15, 2016

Earth-Moon transfer with Sun-perturbation (4-body problem)

Hiten [Belbruno and Miller (1993)]

The Hiten transfer was established in the S-E-M-S/C 4-body problem by considering the Sun-perturbation and by employing the theory of Weak Stability Boundaries. [Belbruno and Miller (1993)]

2

Backgrounds

Hohmann transfer (2-body problem)

The elliptic orbit connecting with the low Earth orbit and the lunar orbit. The two impulsive maneuver are required. [Bate et al. (1971)] E

M

3

[Koon et al. (2001)]

[Conley (1968)]

Coupled PRC3BS

The S-E-M-S/C 4-body problem is approximated to coupled two PRC3BS (S-E-S/C and E-M-S/C systems), and then the transfer is constructed based on the tube dynamics in the coupled system. [Koon et al. (2001)]

Transit orbit

Ws2,X

Wu2,P2

Ws1,P2

Wu1,P1

m2

Forbidden region

L2L1

P1

P2

X

Equation of motion

Energy

q�2 ⌦ q�q = � 1� µ

|q � q1|3(q � q1)�

µ

|q�q2|3(q�q2)

E =

1

2

|q|2 � 1

2

|q|2 � 1� µ

|q � q1|� µ

|q � q2|= const

q = (x, y)T

µ =m2

m1 +m2Mass ratio：

S/C

(1{¹,0)

({¹,0)

q

m1

m2

・PRC3BP and Tube dynamics

L1, L2, L3 saddle × center

Lagrangian points

L4, L5 Stable (S-E-S/C and E-M-S/C systems)

3


[Conley (1968)]

Coupled PRC3BS


Transit orbit

Ws2,X

Wu2,P2

Ws1,P2

Wu1,P1

m2

Forbidden region

L2L1

P1

P2

X

Equation of motion

Energy

q�2 ⌦ q�q = � 1� µ

|q � q1|3(q � q1)�

µ

|q�q2|3(q�q2)

E =

1

2

|q|2 � 1

2

|q|2 � 1� µ

|q � q1|� µ

|q � q2|= const

q = (x, y)T

µ =m2

m1 +m2Mass ratio：

S/C

(1{¹,0)

({¹,0)

q

m1

m2


Lyapunov orbit


Lagrangian points


3


[Conley (1968)]

Coupled PRC3BS


Transit orbit

Ws2,X

Wu2,P2

Ws1,P2

Wu1,P1

m2

Forbidden region

L2L1

P1

P2

X

Equation of motion

Energy

q�2 ⌦ q�q = � 1� µ

|q � q1|3(q � q1)�

µ

|q�q2|3(q�q2)

E =

1

2

|q|2 � 1

2

|q|2 � 1� µ

|q � q1|� µ

|q � q2|= const

q = (x, y)T

µ =m2

m1 +m2Mass ratio：

S/C

(1{¹,0)

({¹,0)

q

m1

m2


Lyapunov orbit


Lagrangian pointsStable manifold


3


[Conley (1968)]

Coupled PRC3BS


Transit orbit

Ws2,X

Wu2,P2

Ws1,P2

Wu1,P1

m2

Forbidden region

L2L1

P1

P2

X

Equation of motion

Energy

q�2 ⌦ q�q = � 1� µ

|q � q1|3(q � q1)�

µ

|q�q2|3(q�q2)

E =

1

2

|q|2 � 1

2

|q|2 � 1� µ

|q � q1|� µ

|q � q2|= const

q = (x, y)T

µ =m2

m1 +m2Mass ratio：

S/C

(1{¹,0)

({¹,0)

q

m1

m2


Lyapunov orbit


Lagrangian points

Unstable manifold

Stable manifold


4

y

v y

Unstable manifold (S-E-S/C)

Stable manifold (E-M-S/C)

Patch point

Boundary condition(departure：low Earth orbit 169km，arrival：low lunar orbit 100km)

Transfer in the S-E rotating frame

U

yv x



Invariant manifold on U

�VP

Coupled PRC3BS


Transfer

Hohmann 3.141 0.838 3.979Coupled PRC3BP [Koon et al. 2001] 3.537 1.989 0.098 5.624

�VE [km/s] �VM [km/s] �VP [km/s] �VTotal

[km/s]

4

y

v y



Patch point

Boundary condition(departure：low Earth orbit 169km，arrival：low lunar orbit 100km)

Transfer in the S-E rotating frame

U

yv x




�VP

How do we find a low energy transfer in the coupled system ?

Coupled PRC3BS


Transfer

Hohmann 3.141 0.838 3.979Coupled PRC3BP [Koon et al. 2001] 3.537 1.989 0.098 5.624


[km/s]

5

LEO:

LLO:

169 [km], 7.087 [km/s]

100 [km], 1.633 [km/s]

Approach to a problem・Use optimization algorithm for a patch point to construct a low energy transfer [Peng et al. (2010)]

・Utilize the tubes (invariant manifolds) near the LEO and LLO to obtain a low energy transfer

∆VE

∆VM

∆VP

LLO

LEO

到着軌道 (E-M-S/C系 )

Patch point

S

E

M

出発軌道 (S-E-S/C系 )

Arrival trajectory (E-M-S/C system)

Departure trajectory (S-E-S/C system)

6

y

x

L2U E

Departure trajectory

LEO

y

x

S ∆VE

E

LEO

Sun

Stable manifold

Departure point

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO ��VE)

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO)

Velocity : increase

Departure trajectory in the S-E-S/C system

Investigate the energy range ( range) such that an orbit is to be a non-transit orbit�VE

＊maneuver 　 uniquely gives�VE ESE

(ESE = �1.50039)

6

y

x

L2U E


LEO

y

x

S ∆VE

E

LEO

Sun

Stable manifold

Departure point

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO ��VE)

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO)

Velocity : increase




ELEO = �1.53501Energy

Energy

Energy : increase

ESE

ESELEO

(ESE = �1.50039)

6

y

x

L2U E


LEO

y

x

S ∆VE

E

LEO

Sun

Initial point of departure trajectory should be outside of the stable manifold

Stable manifold

Departure point

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO ��VE)

(x, y, vx

, v

y

) = (1� µS � rLEO, 0, 0,�vLEO)

Velocity : increase




ELEO = �1.53501Energy

Energy

Energy : increase

ESE

ESELEO

(ESE = �1.50039)

7

∆VE

LEO

x

vy

vx

U


Stable manifold on U

Inside of stable manifold

Stable manifold

Departure

vy

vx

(ESE = �1.50039)

8

∆VE

LEO

x

vy

vx

U


Stable manifold


Outside of stable manifold

Departure

vx

vy

(ESE = �1.50040)

8

∆VE

LEO

x

vy

vx

U


Stable manifold


Outside of stable manifold

Departure

vx

vy

Upper limit of the energy of the departure trajectory (non-transit orbit)

EDMax

(E = �1.50040)ESED

max

(ESE = �1.50040)

9

y

x

ESun

Family of the departure trajectories (non-transit orbits) parametrized by the energy

Energy at the Lagrangian point : EL2= �1.50045L2

Upper limit of the energy : EDMax

= �1.50040


ESE 2 [ESEL

2

, ESED

max

]

ESEL2

ESED

max

10

U L2

x

y

M

Energy range ( range) such that an orbit is to be a transit orbit

Final point is inside of the unstable manifold

∆VM

LLO

x

y

E

M

Arrival point

LLO

(x, y, vx

, v

y

) = (1� µM � rLLO, 0, 0,�vLLO ��VM)

(x, y, vx

, v

y

) = (1� µM � rLLO, 0, 0,�vLLO)

Energy

�VM

Energy ELLO = �2.75466

Velocity and energy : decrease

Earth

Arrival trajectory in the E-M-S/C system

Unstable manifold

Arrival trajectory

＊maneuver 　 uniquely gives�VM EEM

EEM

EEMLLO

(EEM = �1.57962)

11

∆VM

LLO

x

vy

vx

U


Arrival

Unstable manifold

Outside of unstable manifold

UUnstable manifold on(EEM = �1.57962)

12

x

vy

vx

U

∆VM

LLO


Unstable manifold

Arrival

Inside of unstable manifold


12

x

vy

vx

U

∆VM

LLO


Unstable manifold

Arrival

Inside of unstable manifold


Lower limit of the energy of the arrival trajectory (transit orbit)EAMin(E = �1.57961)EEM

Amin

13

y

x

E M

Energy at the Lagrangian point : EL3 = �1.50608L3

Lower limit of the energy : EAMin = �1.57961


Family of the arrival trajectories (transit orbits) parametrized by the energy

EEM 2 [EEMAmin

, EEML3

]

EEMAmin

EEML3

Lunar orbit

E

M µM

x

y U

14

x

vx

vy

✓M 2 [0, 2⇡)

Arrival trajectories

Departure trajectories

✓M = 2.58030Arrival trajectory

LEO-LLO transfer

Family of the departure and arrival trajectoriesin the S-E rotating frame

Sun

(✓M = 2.58030)

＊E-M-S/C system is to be non-autonomous system depending on

✓M：Relative angle

Patch section

Family of the departure and arrival trajectories on

✓M

U

Lunar orbit

E

M µM

x

y U

14

x

vx

vy

✓M 2 [0, 2⇡)




Patch pointzero-maneuver

LEO-LLO transfer


Sun

(✓M = 2.58030)



Patch section


✓M

U

Lunar orbit

E

M µM

x

y U

14

x

vx

vy

✓M 2 [0, 2⇡)




Patch pointzero-maneuver

Optimal transfer for a patch point with zero-maneuver

LEO-LLO transfer


Sun

(✓M = 2.58030)



Patch section


✓M

U

15

y

x

Departure point

Arrival point

Patch point

Lunar orbit

EM

y

x

Patch point

Arrival point

Departure point

EM

Transferin the S-E rotating frame

Sun

Sun

LEO-LLO transfer

Transferin the E-M rotating frame

Transfer

Hohmann 3.141 0.838 3.979Coupled PRC3BP 3.537 1.989 0.098 5.624Proposed approach 3.270 0.642 0 3.912


[km/s]

•We designed the transfer from the low Earth orbit (LEO) to the low lunar orbit (LLO) in the context of the coupled planar restricted 3-body system, namely, the Sun-Earth-spacecraft and Earth-Moon-spacecraft systems.

•We constructed the family of the departure trajectories (non-transit orbits) parametrized by the energy by investigating the tube near the Earth. On the other hand, the family of the arrival trajectories (transit orbits) was obtained.

•We chosen the patch point so that the families of the departure and arrival trajectories are intersected on set section, and then we designed the low energy LEO-LLO transfer. The patch point required the zero maneuver, and thus we optimized the maneuver in patching. Further, the total maneuver is 0.068 [km/s] fewer than the Hohmann transfer.

16

Conclusions

17national geographic

Thanks for your attention !

tube dynamics and low energy trajectory from the …sdross/talks/onozaki-yoshimura-ross...tube...

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