V.G. Petukhov
E-mail: petukhov@mtu-net.ru
Khrunichev State Research and
Production Space Center
2
CONTENTS
INTRODUCTION
1. CONTINUATION METHOD
2. OPTIMAL PLANETARY TRANSFER
VARIABLE SPECIFIC IMPULSE PROBLEM
3. OPTIMAL TRANSFER TO LUNAR ORBIT
VARIABLE SPECIFIC IMPULSE PROBLEM
4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR
ELLIPTIC ORBITS
CONSTANT SPECIFIC IMPULSE PROBLEM
CONCLUSION
V.G. Petukhov. Low Thrust Trajectory Optimization
3
INTRODUCTION
It is presented common methodical approach to computation different problems of
low thrust trajectory optimization. This approach basis is formal reduction of
maximum principle’s two points boundary value problem to the initial value
problem. This reduction is realized by continuation method.
V.G. Petukhov. Low Thrust Trajectory Optimization
INTRODUCTION
Low-thrust trajectory optimization:
T.M. Eneev, V.A. Egorov, V.V. Beletsky, G.B. Efimov,
M.S. Konstantinov, G.G. Fedotov, Yu.A. Zakharov,
Yu.N. Ivanov, V.V. Tokarev, V.N. Lebedev,
V.V. Salmin, S.A. Ishkov, V.V. Vasiliev,
T.N. Edelbaum, F.W. Gobetz, J.P. Marec, N.X. Vinh, K.D. Mease, C.G. Sauer,
C. Kluever, V. Coverstone-Carroll, S.N. Williams, M. Hechler, etc.
Continuation method:
M. Kubicek, T.Y. Na, etc.
V.G. Petukhov. Low Thrust Trajectory Optimization
4
INTRODUCTION
Conventional numerical optimization methods shortcomings
• small region of convergence;
• computational unstability;
• neessity to select initial approximation when it is absent any a-priori information
concerning solution.
These problems partially are connected with optimization problem nature (problems
of optimal solution stability, existance, and bifurcation). But most of numerical
methods introduce own restrictions which are not directly connected with the
mathematical problem properties. So the convergence domain of practically all
numerical methods is essential smaller in comparison with the extremal point
attraction domain in the space of unknown boundary value problem parameters.
Methodical shortcomings are connected with the computational unstability, the
convergence domain boundedness, and (in case of direct methods) the big problem
dimensionality.
V.G. Petukhov. Low Thrust Trajectory Optimization
5
INTRODUCTION
Purpose of new continuation method
“Regularization” of numerical trajectory optimization, i.e. elimination (if possible)
the methodical deffects of numerical optimization. Particularly, the was stated and
solved problem of trajectory optimization using trivial initial approximation (the
coasting along the initial orbit for example).
Applied trajectory optimization problems under consideration
1. Planetary low thrust trajectory optimization
(the variable specific impulse problem);
2. Lunar low thrust trajectory optimization within the frame of restricted problem
of three bodies
(the variable specific impulse problem);
3. Optimal low thrust trajectories between non-coplanar elliptical orbits
(the constant specific impulse problem).
V.G. Petukhov. Low Thrust Trajectory Optimization
6
7
1. CONTINUATION METHOD
Problem: to solve non-linear system
f (z )  0
(1)
with respect to vector z
Let z0 - initial approximation of solution. Then
f ( z0 )  b,
(2)
where b - residuals when z = z0.
Let consider z(), where  is a scalar parameter and equation
f ( z)  (1   )b
(3)
with respect to z(). Obviously, z(1is solution of eq. (1). Let differentiate eq. (2) on  and
solve it with respect to dz/d:
dz
 f z1 ( z)b, z(0)  z0
d
Just after integrating eq. (4) from 0 to 1 we have solution of eq. (1).
Equation (4) is the differential equation of continuation method
(the formal reduction of non-linear system (1) into initial value problem (4)).
V.G. Petukhov. Low Thrust Trajectory Optimization
(4)
CONTINUATION METHOD
8
Application of continuation method to optimal control
boundary value problem
Optimal motion equations
(after principle maximum application):
dx

 Hp , 

dt

dp
  H x 
dt

Boundary conditions (an example):
x(0)  x 0 , x(T ) x k
Boundary value problem parameters and residuals:
z  p(0), f  x(T )  x k
Sensitivity matrix:
fz 
Associated system of optimal motion o.d.e. and
perturbation equations for residuals and sensitivity
matrix calculation:
x(T )
z
dx

 Hp ,

dt

dp

 H x ,

dt
d  x 
x
p 
 H pp
,
   H px
dt  z 
z
z 
d  p 
x
p 
 H xp
    H xx
dt  z 
z
z 
Extended initial conditions:
V.G. Petukhov. Low Thrust Trajectory Optimization
x(0)  x 0 , x(T ) x k ,
x
p
 0,
I
z
z
CONTINUATION METHOD
9
Using continuation method
for low-thrust trajectory optimization problem
Optimal control problem reduction
to the boundary value problem
by maximum principle
CONTINUATION METHOD
Initial
approximation z0
1st version
of o.d.e. right
parts calculation
Initial residuals b calculation
by optimal motion o.d.e. integrating
for given initial approximation z0
of boundary value problem parameters
Continuation method’s o.d.e. integrating
dz
 f z1 ( z)b, z(0)  z0
d
2nd version
of o.d.e. right parts
calculation
with respect to  from 0 to 1
Associated integrating of optimal motion
equations and perturbations equations for
current z() to calculate current residuals f(z,)
and sensitivity matrix fz(z,)
Integrating of optimal motion equations for
current z() to calculate current residuals f(z,)
and for pertubed z() to calculate fz(z,) by finitedifference
Solution
z(1)
V.G. Petukhov. Low Thrust Trajectory Optimization
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2. OPTIMAL PLANETARY TRANSFER
VARIABLE SPECIFIC IMPULSE PROBLEM
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL PLANETARY TRANSFER
11
2.1. TRAJECTORY OPTIMIZATION PROBLEM
1 T 2
Cost function:
a dt (constant power, nuclear electric propulsion)

0
2 2
1 T a
dt (variable power, solar electric propulsion)

2 0 N (x , t )
Equations of motion:
d2x/dt2=x+a
Initial conditions:
x(0)=x0(t0), v(0)=v0(t0)+Ve
Boundary conditions
1) rendezvous:
x(T)=xk(t0+T), v(T)=vk(t0+T)
2) flyby:
x(T)=xk(t0+T)
where x, v - SC position and velocity vectors,  - gravity field force function,
a - thrust acceleration vector, x0, v0 - departure planet position and velocity vectors,
xk, vk - arrival planet position and velocity vectors, V - initial hyperbolic excess of SC velocity, e direction of V, N(x,t) - the current power to the initial one ratio.
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL PLANETARY TRANSFER
12
2.2. OPTIMAL MOTION EQUATIONS
(CONSTANT POWER)
1
dx
H   a Ta  pTx
 pTv x  pTv a
2
dt
a  pv
Hamiltonian:
Optimal control:
Optimal motion equations:
dx
~ 1
H  p Tv p v  p Tx
 p Tv x
2
dt

d 2x



p
,
x
v 

dt 2

d 2p v

2   xx p v . 

dt
Residuals:
x(T ; a 0 , a 0 )  x k 
f 
 (rendezvous)
 v(T ; a 0 , a 0 )  v k 
Optimal Hamiltonian:
x(T ; a 0 , a 0 )  x k  (flyby)
f 

p v (T ; a 0 , a 0 ) 
Boundary value problem parameters
and initial residuals vectors:
a 0 
z  :
a 0 
z  z0
V.G. Petukhov. Low Thrust Trajectory Optimization
f ( z0 )  b
OPTIMAL PLANETARY TRANSFER
2.3. EQUATIONS OF CONTINUATION METHOD
Boundary value problem immersion
into the one-parametric family:
f ( z)  (1   )b
Boundary value problem parameters
initial value and solution:
z  0  z0 , z  1  ~z
Differential equations of continuation
method:
dz
 f z1 ( z)b, z(0)  z0
d
Differential equations for calculation right parts of
continuation method’s differential equations:









p v
d 2  p v  
x
 xx
,
 p 


p v o 
d t 2  p v o  x xx v p v o

p v
d 2  x 
x


,

  xx
p v o p v o
d t 2  p v o 


2
p v 
d  p v  
x
 xx
.
 p 


p v o 
d t 2  p v o  x xx v p v o
d2 x
 x  p v ,
dt2
d2 pv
 xx p v ,
dt2
p v
d 2  x 
x

,
  xx
2 
p v o p v o
d t  p v o 
x(0)  x 0 (0),
p
dx(0)
 v 0 (0)  V v ,
dt
pv
p v p Tv  x (0)
x(0)
x(0)
x (0) V 

 0,
 0,

I  2 ,
 0,
p v0
p v0
p v0
pv 
pv  p v0
p v0 (0)
p (0)
p (0)
p (0)
 I, v0
 0, v0
 0, v0
I
p v0
p v0
p v0
p v0
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
2.4. TRAJECTORY SEQUENCE
WHICH IS CALCULATED BY CONTINUATION METHOD
USING COASTING AS INITIAL APPROXIMATION
5
4
Earth-to-Mars, rendezvous,
launch date June 1, 2000, V= 0 m/s,
T=300 days
1-
3
1 2
coast trajectory
(1= 0)
2-4 - intermediate trajectories
(0 < 2 < 3 < 4 < 1)
5 - final (optimal) trajectory
(5= 1)
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
2.5. NUMERICAL EXAMPLES
OPTIMAL TRAJECTORIES TO MERCURY AND NEAR-EARTH ASTEROIDS
V.G. Petukhov. Low Thrust Trajectory Optimization
15
OPTIMAL PLANETARY TRANSFER
OPTIMAL ORBITAL PLANE ROTATION EXAMPLES
Optimal 90°-rotation
of orbital plane
Optimal 120°-rotation
of orbital plane
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
EXAMPLE: INITIAL HYPERBOLIC EXCESS OF VELOCITY IMPACT
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
EXAMPLE: NUCLEAR (RIGHT) AND SOLAR (LEFT) ELECTRIC PROPULSION
V.G. Petukhov. Low Thrust Trajectory Optimization
18
OPTIMAL PLANETARY TRANSFER
2.7. METHOD OF CONTINUATION WITH RESPECT TO GRAVITY PARAMETER
Reasons of continuation method failure: sensitivity matrix degeneration (bifurcation of optimal solutions)
Mostly bifurcations of optimal
planetary trajectories are
connected with different number
of complete orbits
If angular distance will remain constant during continuation, the
continuation way in the parametric space will not cross boundaries of
different kinds of optimal trajectories. So, the sensitivity matrix will
not degenerate
The purpose of method modification - to fix angular distance of transfer during continuation
Sequence of trajectory calculation using
basic continuation method
Sequence of trajectory calculation using
continuation with respect to gravity parameter
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
Let x0(0), x0(T) - departure planet position when t=0 and t=T;
xk - target planet position when t=T. Let suppose primary gravity
parameter to be linear function of , and let choose initial value of
this gravity parameter 0 using following condition:
1) angular distances of transfer are equal when =0 and =1;
2) When =1 primary gravity parameter equals to its real value (1
for dimesionless equations)
The initial approximation is SC coast motion along departure planet
orbit. Let the initial true anomaly equals to 0 at the start point S,
and the final one equals to k=0+ at the final point K ( is angle
between x0 and projection of xk into the initial orbit plane).
The solution of Kepler equation gives corresponding values of mean anomalies M0 and Mk (M=E-esinE, where
E=2arctg{[(1-e)/(1+e)]0.5tg(/2)} is eccentric anomaly). Mean anomaly is linear function of time at the keplerian
orbit: M=M0+n(t-t0), where n=(0/a3)0.5 is mean motion. Therefore, the condition of angular distance invarianct is
Mk+2 Nrev=nT+M0, where Nrev is number of complete orbits. So initial value of the primary gravity parameter is
0=[( Mk+2 Nrev - M0)/T]2a3,
and current one is
()=0+(1-0) .
The shape and size of orbits should be invariance witn respect to , therefore
v(t, )=()0.5 v(t, 1).
V.G. Petukhov. Low Thrust Trajectory Optimization
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OPTIMAL PLANETARY TRANSFER
21
Equations of motion:
  ()x  p v , p
 v  ()xx p v
x
Boundary conditions:
x(0)  x 0 , x (0)   1/ 2 ( ) v 0 ,
x(T )  x k , x (T )   1/ 2 ( ) v k .
Residuals:
 x (T )  x k

f 

1/ 2
 x ( T )   ( ) v k 
Boundary value problem parameters:
z = (pv(0), dpv(0)/dt)T =  p v o , p v o 
Equation of continuation method:
dz
f 

 fz1 (z) b  , z(0)  z 0
d
 

T
where
d 2  x 
x p v

,
   ( ) xx
2 
z
z
dt  z 
b=
f(z0)
 x (T ) p v o
f z   d
 x(T ) p v o 
dt


d
 x(T ) p v o  
dt
x (T ) p v o
x




f




1
 
  x 
vk 

  2 1/ 2 ( )  
p v 
d 2  p v 
x

,
   ( )   xx p v    xx
2 
z
z 
dt  z 
 x
d 2  x  
x p v
 x   ( ) xx

,

2 
z
z
dt    
p v 
d 2  p v  
x

 xx p v   ( )   xx p v    xx
,

2 
z
z 
dt    
 x
x (0) x (0) x (0)
x (0)
1



 0,

v0 ,
1/ 2
z
z


2 ( ) 
p v (0) p v (0)
p v (0) p v (0) p v (0) p v (0)

 E,



 0.
p v0
p v0
p v0
p v0


V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL PLANETARY TRANSFER
22
Numerical example: Mercury rendezvous
Constant power, launch date January 1, 2001, transfer duration 1200 days
All solutions are obtained using coasting along the Earth orbit as initial approximation
Basic version
of continuation method
Continuation with respect to gravity parameter
5 complete orbits
V.G. Petukhov. Low Thrust Trajectory Optimization
7 complete orbits
OPTIMAL PLANETARY TRANSFER
EXAMPLES: OPTIMAL TRAJECTORIES TO MAJOR PLANETS OF SOLAR SYSTEM
V.G. Petukhov. Low Thrust Trajectory Optimization
23
24
3. OPTIMAL TRANSFER TO LUNAR ORBIT
VARIABLE SPECIFIC IMPULSE PROBLEM
It is considered the transfer of SC using variable specific impulse thruster from a
geocentric orbit into an orbit around the Moon.
The SC trajectory is divided into the 4 arcs:
1) Geocentric spiral untwisting from an initial orbit up to a geocentric intermediate
orbit;
2) L2-rendezvous trajectory;
3) Trajectory from the point L2 of Earth-Moon system to a selenocentric
intermediate orbit;
4) Selenocentric twisting down to a final orbit.
The 1st and 4th arcs can be eliminated if initial and final orbits have high altitude.
Trajectories of 2nd and 3rd arcs are defined by continuation method.
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL TRANSFER TO LUNAR ORBIT
25
VALIDATION OF TRAJECTORY DIVIDING INTO ARCS
Curves of zero velocity
(contours of Jacoby’s integral)
1. Typical duration of hyperbolic
motion within Hill’s sphere of
Moon is ~1 days.
2. Typical velocity increment due to
thrust acceleration is ~10 m/s for 1
day if thrust acceleration is ~0.1
mm/s2.
Hill’s
sphere
Region of
satellite motion
Moon
opening
width
~60000 km
3. Opening width in the L2 vicinity
is ~60000 km for SC relative
velocity 10 m/s on the Hill’s sphere.
to Earth
To capture SC into the Moon
orbit using electric propulsion
(thrust acceleration ~0.1 mm/s2)
SC relative velocity should be not
greater ~10 m/s when distance
from L2 is less ~30000 km.
Region of SC motion for critical Jacoby’s constant
Region of SC motion for SC relative velocity 10 m/s on the Hill’s sphere
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL TRANSFER TO LUNAR ORBIT
26
EARTH_MOON L2 RENDEZVOUS
Model problem of transfer from circular Earth orbit
(altitude 250000 км, inclination 63°, right ascension of ascending node 12°,
lattitude argument 0°; launch date January 5, 2001)
4 complete orbits
5 complete orbits
6 complete orbits
7 complete orbits
0.5
a,
mm/s2
0.0
0.0
t, days
95.0 0.0
t, days
95.0 0.0
t, days
V.G. Petukhov. Low Thrust Trajectory Optimization
95.0 0.0
t, days
95.0
OPTIMAL TRANSFER TO LUNAR ORBIT
27
EARTH-MOON L2 RENDEZVOUS USING MOON GRAVITY ASSISTED MANEUVER
2.5 orbits
Gravity assisted maneuver
7.5 orbits
Initial Moon position
Initial L2 position
Initial orbit
Moon orbit
Final L2 position
Final Moon position
1.0
Thrust acceleration,
mm/s2
Thrust acceleration,
mm/s2
0.5
0.0
0
Т, days
0.0
95
0
V.G. Petukhov. Low Thrust Trajectory Optimization
Т, days
95
OPTIMAL TRANSFER TO LUNAR ORBIT
28
TRANSFER FROM EARTH-MOON L2 INTO CIRCULAR MOON ORBIT
Final orbit: r = 20000 km, i = 90.
Transfer: 2.5 orbits, T = 20 days
Final orbit: r = 30000 km, i = 0.
Transfer: 2.5 orbits, T = 15 days
Final orbit: r = 30000 km, i = 0.
Transfer: 1.5 orbits, T = 10 days
Final
(intermediate)
orbit
Final L2 position
Initial L2 position
Moon
Thrust
acceleration
0.5 mm/s2
0 mm/s2
0
Time, d
10 0
Time, d
15 0
V.G. Petukhov. Low Thrust Trajectory Optimization
Time, d
20
OPTIMAL TRANSFER TO LUNAR ORBIT
TRANSFER FROM EARTH-MOON L2 INTO ELLIPTICAL MOON ORBIT
(i=90°, hp=300 km, ha=10000 km, 10.5 orbits)
Initial L2 position
Final orbit
Moon
Final L2 position
Thrust
acceleration
1 mm/s2
0 mm/s2
0
Time, d
30
V.G. Petukhov. Low Thrust Trajectory Optimization
29
OPTIMAL TRANSFER TO LUNAR ORBIT
30
TRANSFER FROM ELLIPTICAL EARTH ORBIT INTO CIRCULAR MOON ORBIT.
TRAJECTORY ARCS
Geocentric spiral untwisting
Earth-Moon L2 rendezvous
Transfer from Earth-Moon L2
into equatorial 30000-km
circular Moon orbit
Moon
Earth
Earth
0.5 mm/s2
Thrust
acceleration
Thrust
acceleration
0.5 mm/s2
0 mm/s2
0 mm/s2
0
Time, d
95
V.G. Petukhov. Low Thrust Trajectory Optimization
0
Time, d
95
31
4. OPTIMAL MULTI-REVOLUTION TRANSFER
BETWEEN NON-COPLANAR ELLIPTIC ORBITS
CONSTANT SPECIFIC IMPULSE PROBLEM
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
32
Equations of SC motion are written in the equinoctial elements which have not singularty when
eccentricty or inclination is nullified. The optimal control problem is reduced into the two-point
boundary value problem by maximum principle.
This boundary value problem is reduced into the initial value problem by continuation method. It is
necessary to integrare system of optimal motion o.d.e. (P-system) and to calculate partial derivatives of
final state vector of P-system on the initial value of co-state variables to calculate right parts of
continuation method’s o.d.e.
The right parts of the P-system are numerically averaged over true lattitude during the P-system
integration. Partial derivative of final state vector of P-system on the initial value of co-state vector is
calculating using finite differences.
The boundary value problem residual vector are calculated as result of first integration of P-system. 6
additional integrations of P-system is required to calculate sensitivity matrix using finite differences. As
result, the right parts of the continuation method’s o.d.e. are calculated after solving correspoding linear
system.
System of continuation method’s o.d.e. is numerically integrated on continuation parameter  from 0 to
1. As a result, the optimal solution is calculated.
V.G. Petukhov. Low Thrust Trajectory Optimization
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.1. EQUATION OF MOTION
Thrust acceleration components in the orbital reference frame:
a  
P
cos cos
m
ar  
P
sin  cos
m
an  
 - thrust switching function, P - thrust, m - SC mass,  - pitch,  - yaw
P
sin 
m
System of equinoctial elements:
h
p

ex  e cos    e y  e sin    
i
i x  tan cos 
2
i y  tan
i
sin 
2
F     
 - primary gravity parameter; p, e, , , i,  - keplerian elements.
Equation of motion
in the equinoctial
elements:
  1  e x cos F  e y sin F
  i x sin F  i y cos F
~  1  i x2  i y2
w - exhaust velocity
dh
Ph


 h cos  cos ,

dt
m

de x
Ph
 sin F sin  cos    1 cos F  e x cos cos  e y sin  , 

dt
m

de y

Ph
  cos F sin  cos    1 sin F  e x cos cos  e x sin  ,

dt
m


di x
Ph 1~

  cos F sin  ,

dt
m 2


di y
Ph 1~

  sin F sin  ,

dt
m 2

2

dF 
Ph

 3 
  sin  ,
dt h
m


dm
P

  ,
dt
w

Boundary conditions: t = 0: h  h0 , ex  ex 0 , e y  e y 0 , i x  i x 0 , i y  i y 0 , m  m0
t = T: h  hk , ex  exk , e y  e yk , i x  i xk , i y  i yk
V.G. Petukhov. Low Thrust Trajectory Optimization
33
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.2. OPTIMAL CONTROL
T
J  
Cost function:
0
Hamiltonian:
H  
P
dt  min
w
2
P
1  pm    3 pF   P h  A cos cos  Ar sin  cos  An sin  
w
h
m
A  hp h    1 cos F  e x  p ex    1 sin F  e y p ey
Ar   sin F  p ex  cos F  p ey 
1
An    e y p ex  e x p ey   ~ cos F  p ix  sin F  p iy     p F
2
cos 
Optimal control:
sin  
Ar
A  A
Ar
2
2
r
A  A
2
2
r
cos 
sin  
A2  Ar2
A  A  A
An
2
2
r
2
n
A2  Ar2  An2
1, s  0
0, s  0
 
s  
или   1
12
1  pm
h


A2  Ar2  An2 
w
m
12
P
2
Ph 2

H   1  pm   3 pF  
A  Ar2  An2 
w
h
m
Optimal Hamiltonian:
Averaged Hamiltonian does not depends on F, so after averaging dp F   H  0 . So as orbit-to-orbit transfers are
dt
F
considered, the final value F=F(T) is not fixed  pF(T)=0 (transversality condition)
12
 p F  0 it can be missed terms including pF  H   P 1  pm    P A~2  A~r2  A~n2  , where
~  h A ,A
~  h A ,A
~ hA
A


r
r
n
n


w
m

V.G. Petukhov. Low Thrust Trajectory Optimization
34
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.3. EQUATIONS OF OPTIMAL MOTION (P-SYSTEM)
~
~
~ 

A
A
dx H
P ~ 2 ~ 2 ~ 2 1 2  ~ A
~
~

n

  A  Ar  An   A
 An
 An n , 
dt
p
m
p
p
p 



dm H
P

  ,

dt pm
m


~
~
~
dp
H
P ~ 2 ~ 2 ~ 2 1 2  ~ A ~ An ~ An  

  A  Ar  An   A
 An
 An
,
dt
x
m
x
x
x  


dpm
H
P ~2 ~2 ~2 1 2


  2 A  Ar  An  ,

dt
m
m
where
x  h, ex , e y , i x , i y  , p   ph , pex , pey , pix , piy 
T
T
- state and co-state vectors,
~
A
h Ai
i

, i   , r , n.
p  p
~
A
1
i
  Ai
h  
~
A
h Ai
i

i x  i x
~
Ai  A
; i
h  e x
~
A
h Ai
; i 
i x  i x
h
~
A  A
A
h  cos F
h  sin F
 
Ai  i ; i   
Ai  i


e x  e y  

e y
~
A
A 
h  e x sin F  e y cos F
; i  
Ai  i , i   , r , n.
F  

F 

V.G. Petukhov. Low Thrust Trajectory Optimization

;


35
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
A
A
A
A
A
A
 h;     1 cos F  e x ;     1sin F  e y ;       0;
p h
pex
pey
pix piy p F


Ar
A
A
A
A
A
 0; r   sin F ; r   cos F ; r  r  r  0;
p h
pex
pey
pix piy p F
An
A
A
A
A
A
1
1
 0; n  e y ; n  e x ; n  ~ cos F ; n  ~ sin F ; n  .
p h
pex
pey
pix 2
piy 2
p F
A
A
A
 p h ;   cos 2 F  1 p ex  cos F sin F  p ey ;   sin 2 F  1 p ey  cos F sin F  p ex ;
h
e x
e y




A A

 0;
i x
i y
A
 e y cos F  e x sin F cos F    1sin F p ex  e y cos F  e x sin F sin F    1 cos F p ey ;
F



Ar
A
A
 0; r  cos F   pex sin F  pey cos F ; r  sin F   pex sin F  pey cos F ;
h
e x
e y
Ar Ar
A

 0; r   e x sin F  e y cos F  pex sin F  pey cos F     pex cos F  pey sin F ;
i x
i y
F
An
A
A
A
A
 0; n  pey  p F cos F ; n  pex  p F sin F ; n  n  0;
h
e x
e y
i x
i y
An
1
 i x cos F  i y sin F e x pey  e y pex   ~  piy cos F  pix sin F  
F
2
 e y cos F  e x sin F p F  i x cos F  i y sin F p F .
V.G. Petukhov. Low Thrust Trajectory Optimization

36
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.4. BOUNDARY VALUE PROBLEM
Within the fixed-time problem equation of residuals is following:
This equation should be solved with respect to unknown initial value of co-state
vector p(0), pm(0).
Within the minimum time problem   1 and equations for m and pm are
eliminated by substitusion expression m = m0 - (P/w) t into other equations.
Equation of residuals is following:
This equation should be solved with respect to unknown initial value of co-state
vector p(0) and transfer duration T.
Continuation method’s equation:
 h(T )  hk 


 e x (T )  e xk 
 e (T )  e 
y
yk
0
f 
 i x (T )  i xk 


 i y (T )  i yk 
 p (T )

 m

 h(T )  hk 


 e x (T )  e xk 
 e (T )  e 
y
yk 
f 
0
 i x (T )  i xk 


 i y (T )  i yk 
 H (T ) T 


 ph 


 p ex 
p 
1
ey 
dz
 f 
    b, where z  
(minimum time) or z=p (fixed time);
 pix 
d
 z 


 piy 
T 


b=f(z0) - residual vector for initial z (when =0). The boundary value problem is solved by integration of continuation
method’s equation on  from 0 to 1. Partial derivatives of residual vector f on vector z and linear system solving for
computation right parts of o.d.e. are processed numerically.
V.G. Petukhov. Low Thrust Trajectory Optimization
37
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.5. DETAILS OF BOUNDARY VALUE PROBLEM SOLVING
Boundary value problem is solved by continuation method.
The averaged with respect to true lattitude equations of optimal motion are used to calculate residuals f.
These equations have singularity when co-state vector p=0, so it is impossible to use zero initial co-state
vector (coast motion) as initial approximation.
Within the minimum time problem the following initial approximation was used: ph(0)=1 if the final semimajor axis greater than the semi-major axis of initial orbit and ph(0)=-1 otherwise. The rest vector p
components were picked out equal to 0 and the initial approximation of transfer duration was T|=0=1
(dimensionless). Using this initial approximation there were found the minimum-time transfers to GEO
from the elliptical transfer orbits having inclination 0°-75° and apogee altitude 10000-120000 km. If initial
apogee altitude was not match with this range, the solution for a transfer from close initial orbit was used
as the initial approximation.
It is used numerical averaging the equations of optimal motion on the true lattitude F during these
equations integration.
The partial derivatives of residuals f with respect to p(0), T, which are necessary for continuation method,
are processed numerically using finite differences.
So, there are used numerical integration of numerically averaged equations of optimal motion and
numerical differentiating of residuals to calculate right parts of continuation method’s o.d.e.
V.G. Petukhov. Low Thrust Trajectory Optimization
38
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.6. OPTIMAL SOLUTION IN NON-AVERAGED MOTION
The real and averaged evolutions of orbital motion are close each to other due to the
relatively low thrust acceleration level.
To check accuracy of optimal averaged solution, the obtained optimal p(0) and T were
used for numerical integration of non-averaged equations of motion. The initial value of
true lattitude F was chosen arbitrary (the perigee or apogee values mostly). The initial
value of pF was equals to 0 (see note above).
The optimal thrust steering and insertion errors were calculated as result of numerical
integration of the non-averaged equations. The relative errors due to averaging did not
exceed 0.1% for transfer from an elliptical orbit to GEO when thrust acceleration was
0.1-0.5 mm/s2.
An optimal thrust steering examples are presented below.
V.G. Petukhov. Low Thrust Trajectory Optimization
39
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.7. OPTIMAL ORBITAL EVOLUTION
AND OPTIMAL THRUST STEERING
(MINIMUM-TIME PROBLEM)
Р аPerigee
д и у с п еdistance
ригея
80000
Distance, km
Ра сстояние
, км
Orbital evolution for suboptimal apogee
altitude of initial orbit
(ha = 30000 km, i = 75°)
Р аApogee
д и у с а пdistance
огея
70000
Semi-major axis
Б оль ш ая п олу ос ь
60000
50000
40000
30000
20000
10000
0
0
50
100
150
В р е мdays
я , сут
Time,
Inclination,
На клоне
ние , deg
гра дусы
80
1. Average apogee, semi-major axis, and eccentricity have
maximum during transfer.
2. Perigee distance increases monotonously.
70
60
50
40
30
20
10
0
0
50
100
150
ВTime,
р е м яdays
, сут
0,8
Eccentricity
Эксце
нтрисите т
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
50
100
150
В р е м яdays
, сут
Time,
V.G. Petukhov. Low Thrust Trajectory Optimization
40
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
Accelerationbraking
Brakingacceleration
Acceleration
Optimal thrust steering for suboptimal
apogee altitude of initial orbit
(ha = 30000 km, i = 75°)
Yaw, deg градусы
Угол рысканья,
90
60
30
0
-30
-60
-90
0
20
40
60
80
100
120
140
160
120
140
160
120
140
160
Тангаж,
deg
Pitch, градусы
Время,
сутки
Time,
days
180
150
120
90
60
30
0
-30
-60
-90
-120
-150
-180
0
20
40
60
80
100
deg
of attack,
Angle атаки,
Угол
градусы
Время,
сутки
Time,
days
180
150
120
90
60
30
0
-30
-60
-90
-120
-150
-180
0
20
40
60
80
100
Время,
Time,
daysсут
V.G. Petukhov. Low Thrust Trajectory Optimization
41
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
pitch
тангаж
angle
of attack
угол
атаки
path angle угол
траекторный
180
150
120
90
60
30
0
-30
-60
-90
-120
-150
-180
90
Yaw, deg градусы
Угол рысканья,
Угол,
градусы
deg
Angle,
OPTIMAL THRUST STEERING
60
30
0
-30
-60
-90
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
Time,
daysсут
Время,
тангаж
pitch
angle
of attack
угол
атаки
path angle угол
траекторный
deg
Angle,градусы
Угол,
60
30
0
-30
-60
-90
1
60
30
0
-30
-60
-90
80
80.2 80.4 80.6 80.8
81
81.2 81.4 81.6 81.8
82
80
Время,
сут
Time, days
180
150
120
90
60
30
0
-30
-60
-90
-120
-150
-180
141 141.2 141.4 141.6
pitch
тангаж
angle
of attack
угол
атаки
path angle угол
траекторный
141.8 142 142.2 142.4 142.6 142.8 143
Время,
сут
Time,
days
80.2
80.4
80.6
80.8
81
81.2
81.4
81.6
81.8
82
Время,
Time,
daysсут
90
Yaw, deg градусы
Угол рысканья,
deg
Angle,
Угол, градусы
0.8
90
Yaw, deg градусы
Угол рысканья,
90
0.6
Время,
сут
Time, days
60
30
0
-30
-60
-90
141 141.2 141.4 141.6 141.8 142 142.2 142.4 142.6 142.8 143
Время,
Time,
daysсут
V.G. Petukhov. Low Thrust Trajectory Optimization
42
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
43
1 4t=141
1 - е dс у т к и
180
150
120
90
60
30
0
-3 0
-6 0
-9 0
-1 2 0
-1 5 0
-1 8 0
8 0t=80
- е сd у т к и
Угол
атаки,
градусы
deg
of attack,
Angle
deg
Pitch, градусы
Тангаж,
OPTIMAL THRUST STEERING
2 -t=2
е сdу т к и
0
30
60
90
120 150 180 210 240 270 300 330 360
Yaw,, deg
Рысканье
градусы
deg
И с ти н н аTrue
я а нanomaly,
ом алия
, г р а д ус ы
80
1t=141
4 1 - еdс у т к и
70
8t=80
0 - е dс у т к и
60
1 4t=141
1 - е dс у т к и
180
150
120
90
60
30
0
-3 0
-6 0
-9 0
-1 2 0
-1 5 0
-1 8 0
8 0t=80
- е сd у т к и
2 -t=2
е сdу т к и
0
30
60
90
40
30
20
10
0
-1 0
-2 0
-3 0
0
30
60
90
120
150
180 210
240
270
210 240 270 300 330 360
И с ти н н аTrue
я а нanomaly,
о м а л и deg
я , г р а д ус ы
2t=2
- е dс у т к и
50
120 150 180
300 330
360
И с ти н н а яTrue
а н оanomaly,
м а л и я ,deg
г р а д ус ы
V.G. Petukhov. Low Thrust Trajectory Optimization
Distance, km
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
150000
140000
130000
120000
110000
100000
90000
80000
70000
60000
50000
40000
30000
20000
10000
0
Orbital evolution
and optimal thrust steering
for optimal apogee altitude
of initial orbit
(ha = 140000 км, i = 65°)
rp
ra
a
0
10
20
30
40
50
60
70
80
90
100
44
110
t, days
Perigee & apogee distance and semi-major axis
1
180
0.9
150
0.8
120
90
yaw
60
0.6
Angle °
Eccentricity
0.7
pitch
0.5
0.4
30
0
-30
-60
0.3
-90
0.2
-120
-150
0.1
0
-180
0
10
20
30
40
50
60
70
80
90
100
110
0
10
20
30
40
t, days
50
60
t, days
Eccentricity
Eccentricity
V.G. Petukhov. Low Thrust Trajectory Optimization
70
80
90
100
110
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
250000
Orbital evolution
and optimal thrust steering for
superoptimal apogee altitude
of initial orbit
(ha = 240000 км, i = 65°)
rp
ra
200000
a
Distance, km
45
150000
100000
50000
0
0
10
20
30
40
50
60
70
80
90
100
110
120
t, days
Perigee & apogee distance and semi-major axis
Braking
1.00
0.90
0.80
0.60
Angle °
Eccentricity
0.70
0.50
0.40
0.30
0.20
0.10
0.00
0
10
20
30
40
50
60
70
80
90
100
110
120
180
150
120
90
60
30
0
-30
-60
-90
-120
-150
-180
Braking-acceleration
pitch
yaw
0
10
20
30
40
50
60
t, days
t, days
Eccentricity
Eccentricity
V.G. Petukhov. Low Thrust Trajectory Optimization
70
80
90
100 110 120
OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS
4.8. OPTIMIZATION OF TRANSFER FROM ELLIPTIC ORBIT TO GEO
Initial
inclination °
Initial perigee altitude 250 km,
SC mass in the GEO 450 kg, thrust 0.166 N, specific impulse 1500 s
Transfer duration, days
Initial apogee altitude, thousands km
i0=75°
i0=51.3°
i0=65°
i0=0°
Initial apogee altitude, thousands km
V.G. Petukhov. Low Thrust Trajectory Optimization
47
48
CONCLUSION
The developed continuation method demonstrated extremely effectiveness for
variable specific impulse problem. The combination of two continuation versions
(basic continuation method and continuation with respect to gravity parameter)
allows to process planetary mission analysis fast and exhaustevely.
The L2-ended low thrust trajectories were optimized using the continuation
method. These solutions were used to construct quasioptimal trajectories between
Earth and Moon orbits.
The version of continuation method allows to carry out full-scale analysis of the
low-thrust mission to GEO from the inclined elliptical transfer orbit.
So, the continuation method performances make this method an effective and
useful tool for analysis the wide range of electric propulsion mission
V.G. Petukhov. Low Thrust Trajectory Optimization