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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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