Yuan Yuan RenRenDepartament de Matemtica Aplicada I

Universitat Politcnica de Catalunya

LowLow--thrust Trajectory thrust Trajectory OptimizationOptimization

2

Index

The Action of Trajectory Optimization

Calculus of Variations & Pontryagin’s Minimum Principle

Dynamical Models

Numerical Methods

Some Examples

3

Tsiolkovsky formula

Tsiolkovsky Formula

0lnk

MV uM

Δ =

is the initial total mass, including propellant in kg0M

is the final total mass in kgkM

is the effective exhaust velocity in m/su 0spu I g=

is the delta-v in m/sVΔ

4

Chemical rocket engine

Chemical rocket engine used on carrier rocket

Thrust: 10-50tons, Specific Impulse: 300s

Chemical rocket engine used on satellite

Thrust: 490 N, Specific Impulse: 290s

5

Low-thrust engine

6.9 kW, LM 4 40 cm Ion Thruster 1.1 - 6.9 kW, specific impulse 2210 – 4100 s, thrust 50 - 237 mN.

Busek 600 W Hall Thruster Cluster

6

Different design methods

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

M0/Mk

Δ V

(km

/s)

1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

M0/Mk

Δ V

(km

/s)

Chemical rocket Isp=300s Low-thrust engine Isp=4100s

The orbit maneuver needs a long time. So the Impulse Assumption can not be used in the low-thrust orbit design and optimization. We need some new methods to deal with this kind of problem.

The orbit maneuvers are assumed as Impulse of the Velocity. The whole trajectory is consisted by a series of unpowered arcs, which are connected with each other by these ‘impulses’.

Orbit transfer using low-thrust engine:

Orbit transfer using chemical rocket:

7

Functional and its variations

Suppose that x is a continuous function of t defined in the interval [t0, tf ] and

0

( ) ( )ft

tJ x x t dt= ∫

J(x) is a functional of the function x(t). Roughly speaking, the functional is the function of function. The variation plays the same role in determining extreme values of functional as the differential does in finding maxima and minima of functions

The extremum of function The extremum of functional

8

Necessary conditions for optimal control (I)

There is a dynamical equation of a system.( ) ( ( ), ( ), )t t t t=x f x u

The problem is that to find an admissible control u* that causes the system to follow an admissible trajectory x* that minimizes the performance index:

0

( ) ( ( ), ) ( ( ), ( ), )ft

f f tJ t t L t t t dt= Φ + ∫u x x u

We shall initially assume that the admissible state and control regions are not bounded, and that the initial conditions and the initial time are specified.0 0( )t =x x 0t

If is a differentiable function, we can writeΦ

00 0( ( ), ) [ ( ( ), )] ( ( ), )ft

f f t

dt t t t dt t tdt

Φ = Φ +Φ∫x x x

9

Necessary conditions for optimal control (II)

So that the performance index can be expressed as

00 0( ) { ( ( ), ( ), ) [ ( ( ), )] ( ( )} , )ft

t

dJ L t t t t t t tdtdt

= +Φ+ Φ∫u x u x x

Since and are fixed, the minimization does not affect the term, and using the chain rule of differentiation, we find that becomes

0( )tx 0t0 0( ( ), )t tΦ x

( )J u

To include the differential equation constraints, we form the augmented functional

0

( ) ( ( ), ( ), ) ( ( ), ) ( ) ( ( ), )fT

t

tJ L t t t t t t t t dt

t⎧ ⎫∂Φ ∂Φ⎪ ⎪⎡ ⎤= + +⎨ ⎬⎢ ⎥∂ ∂⎣ ⎦⎪ ⎪⎩ ⎭

∫u x u x x xx

[ ]}0

( )

( ) ( ( ), ( ), ) ( ( ), ) ( ) ( ( ), )

( ( ), ( ), ) ( )

fT

t

a t

T t

J L t t t t t t t tt

t tt t t d

⎧ ∂Φ ∂Φ⎪ ⎡ ⎤= + +⎨ ⎢ ⎥∂ ∂⎣ ⎦⎩+

−

⎪∫u x u x x x

x

λ f x u x

10

Necessary conditions for optimal control (III)

Let’s define

[ ]

( ( ), ( ), ( ), ( ), ) ( ( ), ( ), ) ( ( ), ) ( )

( ( ), ) ( ) ( ( ), ( ), ) ( )

T

a

T

L t t t t t L t t t t t t

t t t t t t tt

∂Φ⎡ ⎤= + +⎢ ⎥∂⎣ ⎦∂Φ

+ −∂

x x u λ x u x xx

x λ f x u x

We shall assume that the end points at can be specifiedor free. To determine the variation of , we introduce the variations and .

ft t=aJ

, , ,δ δ δ δx x u λ ftδ

0

*( ) 0 ( )

( ) ( ) ( )

ff f

f

a aa f a f ft

t t

t a a a at

L LJ L t t

L d L L Lt t t dtdt

δ δ δ

δ δ δ

⎡ ⎤∂ ∂= = + − +⎢ ⎥

∂ ∂⎢ ⎥⎣ ⎦⎧ ∂ ∂ ∂ ∂ ⎫⎡ ⎤− + +⎨ ⎬⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦⎩ ⎭

∫

u x xx x

x u λx x u λ

Notice that the above result is obtained because and do not appear in .

( )tu ( )tλ

aL

11

Necessary conditions for optimal control (IV)

If it assumed that the second partial derivatives are continuous, the order of differentiation can be interchanged. In the integral term we have, then,

[ ]0

( ) ( ) ( ) (( ) ( ) ( ))ft

tt t t dL d Lt t t t

dttδ δ δ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤+ + + −⎢

⎧ ⎫+ +⎥ ⎢ ⎥∂ ∂ ∂ ∂

⎬⎦⎩⎣

⎨⎭⎦ ⎣∫

f fλ λ λ f xx x u u

x u λ

The integral must vanish on an extremal regardless of theboundary conditions. If we define , we can observe that the constraints

TH L= + λ f

* *( ( ), ( ), )

0

t t tH

H

⎧ =⎪⎨ ∂

= −⎪ ∂⎩∂

=∂

x f x u

λx

u

12

Necessary conditions for optimal control (V)

There are still the terms outside the integral to deal with; since the variation must be zero, we have

* *( ( ), ) ( 0) ( ( ), )f

f f f fftf ft t t H t t tt

δ δ+∂Φ ∂Φ⎡ ⎤ ⎡ ⎤− +⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦

=x λx

x x

So we obtain the boundary condition.*

*

( ( ), )

( ) ( ( ), )

ff ft

f f f

H t tt

t t t

∂Φ= −

∂∂Φ

=∂

x

λ xx

13

Boundary conditions

Fixed time and end point Fixed time, free end point

Free time and fixed end point Free time and free end point

Final states on the moving point

14

Boundary conditions

15

Dynamical Equations of Classic Orbital Elements

( )

( )

( )

( ) ( )

( ) ( )

2 2

r t

r t

n

n

r t n

r t

2 sin 2

1 1sin cos

cos

sinsin

sin sin coscossin

1 cos 2 sin

a e a pa f fh hr

e p f p r re fh hr

i fh

rf

hp r r ip f f f

h h h

M n p er f p r f

e e

ea h

i

i

υ

υ υ

ω υ

ω υ

υ ω υυω

υ υ

⎧= +⎪

⎪⎪ = + + +⎡ ⎤⎣ ⎦⎪⎪

+⎪ =⎪⎪⎨

+⎪Ω =⎪⎪

+ +⎪ = − + −⎪⎪⎪ = + − − +⎡ ⎤⎣ ⎦⎪⎩

Dynamical equations of classic orbital elements can be written as follow:

We can find that when ‘i’ or ‘e’ equal to zero, the dynamical equations will become singularity. In low-thrust transfer, the orbital elements is changed continuous, so the singularity could hardly be avoided.

IZ

IY

x

y

z

T

IX

16

Modified equinoctial elements

In order to avoid the singularity, we need to use a set of no singularity elements. The modified equinoctial elements is a good choice. The modified equinoctial elements and its dynamical equations are defined as follows:

( )( )( )

( )( )

21

cos

sin

tan / 2 cos

tan / 2 sin

p a e

f e

g e

h i

k iL

ω

ω

ω υ

= −

= +Ω

= +Ω

= Ω

= Ω

= Ω+ +

( ) ( )

( ) ( )

( )

t

t nr

t nr

2n

2n

2

n

2

sin 1 cos sin cos

cos 1 sin sin cos

cos2

sin2

1 sin cos

p pp fw

f g fpf f L w L f h L k Lw w

f f fpg f L w L g h L k Lw w

s fph Lw

s fpk Lw

w pL p h L k L fp w

μ

μ

μ

μ

μ

μμ

⎧=⎪

⎪⎪ ⋅⎧ ⎫⎪ = + + + − −⎡ ⎤⎨ ⎬⎣ ⎦⎪ ⎭⎩⎪

⋅⎧ ⎫⎪ = − + + + + −⎡ ⎤⎨ ⎬⎣ ⎦⎪ ⎭⎩⎪⎨⎪ =⎪⎪⎪

=⎪⎪⎪ ⎛ ⎞⎪ = + −⎜ ⎟⎪ ⎝ ⎠⎩

1 cos sinw f L g L= + +2 2 21s h k= + +

Where

17

Necessary conditions for this modelThe dynamical equations can be describe in matrix form.

6 3 p 6 1

0 sp

Tm

Tmg I

× ×

⎧ ⎛ ⎞= + +⎜ ⎟⎪ ⎝ ⎠⎪⎨⎪ = −⎪⎩

x B α f D

[ ] [ ]

[ ] [ ]

( )

2

2

20 0

1sin ( 1)cos sin cos

1cos ( 1)sin sin cos

0 0 cos2

0 0 sin2

10 0 sin cos

p pw

p p p gL w L f h L k Lw w

p p p fL w L g h L k Lw w

p s Lw

p s Lw

p h L k Lw

μ

μ μ μ

μ μ μ

μ

μ

μ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+ + − −⎢ ⎥⎢ ⎥⎢ ⎥− + + −⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

−⎢ ⎥⎣ ⎦

B

[ ]0 0 0 0 0 Td=D2

wd pp

μ⎛ ⎞

= ⎜ ⎟⎝ ⎠

- specific impulse of the thruster

- acceleration of gravity at sea levelspI

0g

The Hamiltonian can be defined as:

( )TL m

sp 0

1T T TH dm I g

λ λ γ= + − + −λ Bα α α

18

Necessary conditions for this modelBy using the control equation, we can obtain the optimal control:

( )** 0

TT

T

H∂= → =

∂

λ Bα

α λ B

By using Pontryagin minimal principle, the Hamiltonian should always be minimized. So we can get a switch function.

T*T m m

0 sp 0 sp

HT m g I m g I

λ λ∂Γ = = − = −

∂

λ Bαλ B

max

max

0 0 0

0 0

TT T

T T

= Γ >⎧⎪ = Γ <⎨⎪ < < Γ =⎩

pp

2 2

T TL L

T Tm

H T dm

H T Tm m m

λ λ

λ

⎧ ∂⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= − = − − + − −⎪ ⎜ ⎟⎜ ⎟⎪ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎨∂⎪ = − = =⎪ ∂⎩

fB Bλ λ α λ f Bx x x x x

λ Bα λ B

The co-state equations are determined as follow:

( )f f

Tf

t t t t

t φ ψ

= =

∂ ∂= +∂ ∂

λ γx x

f

T

0t t

Ht tφ ψ

=

⎡ ⎤∂ ∂⎛ ⎞+ + =⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎣ ⎦γ

The boundary condition:

19

Optimal trajectory in three-body model

0 sp

U Tm

Tmg I

⎧⎪ =⎪⎪ ∂

= + +⎨ ∂⎪⎪ = −⎪⎩

r v

v Mv αr

( )T T Tr v m

sp 0

1U T THm I g

λ γ∂⎛ ⎞= + + + − + −⎜ ⎟∂⎝ ⎠λ v λ Mv α α α

r

* v

v

=λαλ

T *vv m m

0 sp 0 sp

HT m g I m g I

λ λ∂Γ = = − = −

∂λλ α

max

max

0 0 0

0 0

TT T

T T

= Γ >⎧⎪ = Γ <⎨⎪ < < Γ =⎩

Tr v 2

Tv r v

m v 2

H U

H

H Tm m

λ

∂ ∂⎧ = − = −⎪ ∂ ∂⎪∂⎪ = − = − −⎨ ∂⎪∂⎪ = − =⎪ ∂⎩

λ λr r

λ λ λ Mv

λ

( )f f

Tf

t t t t

t φ ψ

= =

∂ ∂= +∂ ∂

λ γx x

f

T

0t t

Ht tφ ψ

=

⎡ ⎤∂ ∂⎛ ⎞+ + =⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎣ ⎦γ

Conjugate equationsDynamical equations Direction of thrust

Switch function

Hamiltonian

Transversality condition

20

Numerical methods

Direct Method

Indirect Method

Hybrid Method

Solve the Two-point Boundary-value Problem.

Transform the optimal control problem into parameter optimization by discretization.

Disadvantage: Difficult to converge, Difficult to give reasonable a Initial guesses

Advantage: the solution is smooth and accurate

Disadvantage: Time consuming, The solution is not smooth

Advantage: Easy to converge than indirect method, Some other optimization parameters can be added

Neglect the transversality condition, adjust the co-states by using parameter optimization method (SQP)

It is a tradeoff method between indirect and direct method.

21

Hybrid method

* *( ( ), ( ), )

0

t t tH

H

⎧ =⎪⎨ ∂

= −⎪ ∂⎩∂

=∂

x f x u

λx

u

±∞

0t 1t 2t 3t 4t nt

0x1x

2x 3x

1x′2x′

3x′4x′

0λ1λ

2λ3λ

1λ′2λ′

3λ′4λ′

X

T

at

*

*

( ( ), )

( ) ( ( ), )

ff ft

f f f

H t tt

t t t

∂Φ= −

∂∂Φ

=∂

x

λ xx

Transversality condition

0 0( )( )f f

tt

==

x xx x

Boundary condition

1. Neglect the transversality condition;2. Discretization of the states and the co-states;(optional)3. Using constrained parameter optimization method to

minimize the performance index.

Sequential Quadratic Programming (SQP).Software package named SNOPT.Genetic Algorithm (GA), Simulated annealing (SA)

0

( ) ( ( ), ) ( ( ), ( ), )ft

f f tJ t t L t t t dt= Φ + ∫u x x u

Performance Index

Differential equations

22

Orbit transfer from LEO to GEO

TOF p f g h k L m[ , , , , , , , ]t λ λ λ λ λ λ λ=Z

+x+y+z

0 100 200 300 400 500 600 700-80

-60

-40

-20

0

20

40

60

Optimization Parameter:

Constraints:

( ) ( )T TOF GEO 0, , , , ,C t p f g h kζ ζ ζ= − = =

TOFJ t=

Performance Index:

Transfer trajectory

Thruster angles in orbit frame

Green: yaw thruster angle

Blue: pitch thruster angle

23

Low-thrust transfer from parking orbit to halo orbit

11.002

1.0041.006

1.0081.01

-4

-2

0

2

4

10-3

-505

x 10-4

X(AU)Y(AU)

Z(A

U)

L2

稳定流形入口点

Halo轨道入轨点

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

-5

-4

-3

-2

-1

0

1

2

3

4

5

x 10-3

X(AU)

Y(A

U)

L2

稳定流形入口点

Halo轨道入轨点

-5 0 5x 10-3

-1

0

1x 10-3

Y(AU)

Z(A

U)

稳定流形入口点

Halo轨道入轨点

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012-1

0

1x 10-3

X(AU)

Z(A

U)

稳定流形入口点

Halo轨道入轨点

L2

T p f g h k L m p c[ , , , , , , , , , ]t t tλ λ λ λ λ λ λ=ZOptimization Parameter:

Constraints: BACK T LEO( ) 0, ( , , , , )C t p f g h kζ ζ ζ= − = =

TJ t=Performance Index:

24

Direct transfer from the Earth to the Venus

5050

50

5050

50

50

50

100

100

100

100

100

100100

100

100

100

100

200

200

200

200

200

200

200

200

200

200

200

3 00

300

300

300

300

300

300

300

300

300

300

400

400

400

400

400

500

500

500

发射时间(2010年11月1日+N天)

飞行时

间(天

)

0 100 200 300 400 500 600 700 800 90070

80

90

100

110

120

130

140

150

局部极小点1

局部极小点2 L A p f g h k L m[ , , , , , , , , ]t t λ λ λ λ λ λ λ=Z

( ) ( ) ( )T A Venus A 0, , , , , ,C t t p f g h k Lζ ζ ζ= − = =

A LJ t t= −

-1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

-1-0.8

-0.6-0.4

-0.20

0.20.4

0.60.8

1

-0.15

-0.1

-0.05

0

0.05

X(AU)

Y(AU)

Z(A

U)

地球逃逸金星俘获

太阳金星轨道

地球轨道

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X(AU)

Y(A

U)

地球逃逸

金星俘获

太阳

金星轨道

地球轨道

0 20 40 60 80 100 120 140 160 180-200

-150

-100

-50

0

50

100

150

时间(天)

控制

角(d

eg)

俯仰控制角

偏航控制角

Launch opportunity searchLaunch opportunity search

Trajectory optimizationTrajectory optimization

25

low-thrust IPS transfer

Low-thrust & Interplanetary SuperHighway(IPS) have the same aim ---- Increase the percentage of payload

Low-thrust ---- Increase the availability factor of working substance

IPS ---- Decrease the total delta V

Orbit structure of the low-thrust IPS transfer

26

One Thrust Arc IPS transfer

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

X(AU)

Y(AU)

Z(A

U) 地球轨道

金星轨道

太阳

脱离地球L1点

进入金星L2点

WsL2Wu

L1

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X(AU)

Y(A

U)

地球轨道

金星轨道

太阳

脱离地球L1点

进入金星L2点Ws

L2

WuL1

0 50 100 1500.7

0.8

0.9

1

p

0 50 100 1500

0.1

0.2

f

0 50 100 150-0.05

00.05

0.1

g

0 50 100 150-5

0

5

10x 10-3

h

0 50 100 150-0.01

00.010.020.03

k

时间(天)0 50 100 150

3

4567

L

时间(天)

0 50 100 150-10

0

10

20

λ p

0 50 100 150-10

-5

0

5

λ f

0 50 100 150-5

0

5

10

15λ g

0 50 100 1500.2

0.3

0.4

0.5

λ h

0 50 100 150-1.9

-1.85

-1.8

-1.75

λ k

时间(天)0 50 100 150

3

3.5

4

4.5

λ L

时间(天)

0 50 100 150

0.8

0.9

1

M

时间(天)0 50 100 150

-2

-1

0

λ M

时间(天)

L A wu ws p f g h k L m L A[ , , , , , , , , , , , , ]t t t t v vλ λ λ λ λ λ λ δ δ=Zl u l u

L L L A A A, , ,v v v v v vδ δ δ δ δ δ⎡ ⎤ ⎡ ⎤∈ ∈⎣ ⎦ ⎣ ⎦( )ThrustArc A L wu ws= 24 3600 / TU 0T t t t t− × × − − ≥

( ) ( ) ( )ThrustArc c ws c 0, , , , , ,C t t p f g h k Lζ ζ ζ= − = =

ThrustArcJ T=

27

Thrust-coast-thrust IPS transfer

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

X(AU)

Y(AU)

Z(A

U)

太阳

金星轨道

地球轨道

脱离地球L1点进入金星L2点

WL1u WL2

s

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

X(AU)

Y(A

U) 太阳

金星轨道

地球轨道

脱离地球L1点进入金星L2点

WL1u

WL2s

1 1 1 1 1 1 1L A wu ws p f g h k L m

2 2 2 2 2 2 2p f g h k L m L A 1 2

[ , , , , , , , , , , ,

, , , , , , , , , , ]

t t t t

v v t t

λ λ λ λ λ λ λ

λ λ λ λ λ λ λ δ δ

=Z

[ ) [ )l u l uL L L A A A 1 2, , , , 0,1 , 0,1v v v v v v t tδ δ δ δ δ δ⎡ ⎤ ⎡ ⎤∈ ∈ ∈ ∈⎣ ⎦ ⎣ ⎦

( )TCT A L wu ws= 24 3600 / TU 0T t t t t− × × − − ≥

1 2 1t t+ ≤

( ) ( ) ( )TCT c ws c 0, , , , , ,C t t p f g h k Lζ ζ ζ= − = =

( ) ( )A L wu ws 1 224 3600 / TUJ t t t t t t= − × × − − × +⎡ ⎤⎣ ⎦

28

Comparison between the different scenarios

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