team 36: viacheslav petukhov (riame), roman elnikov (mai

2

Problem statement

Spacecrafts: mother ship (ms) and three probes (prob).

0

0 0 0 0

6 km s, MJD 59215 MJD 62867 01.01.202100 : 00 UT, 31.12.2030 24 : 00 UT ,

24000 kg, 3 , ,

6000 kg, 12000 kg, 800 kg, 1200 kg,

900 s,

ms ms prob prob prob prob

dry dry dry dry

ms ms prob prob

dry dry dry dry

ms p

sp sp

v t

m m m m m m m m

m m m m

I I

0 0

3000 s, 0.3N,

12 years, 6 years, 30 days,

rob prob

prob prob prob prob

f f de rv

T

t t t t t t

3

1 1

, ' ( ) .ast

j

N

i j f prob

i j

J J m

16256,astN

3

Sort asteroids

16256 asteroids

Finding pairs of the asteroids at the Close Approach Events:

4

CAE, ,i j na a t

Sort asteroids

1) search “nearby” orbits:

2) search of the close approach time intervals

T

* * * *

0 0 0 90 ,i i k j j kt t t t r r r r

*

j ktr

*

0j tr

*

i ktr

*

0i tr

3) search time moment of the close approach event

T T* * * * * * * * * *

0 0 0 0 CAE 0

T T

CAE CAE CAE CAE CAE CAE CAE CAE

CAE CAE max CAE CAE max

0 ; ,

0,

, .

i j i j i k j k i k j k k

i j i j i j i j

i j i j

t t t t t t t t t t t

t t t t t t t t

t t r t t v

r r v v r r v v

r r v v r r v v

r r v v

ast ast0,1,... , 0,1,... ,i N j N

max, min max ,i ji j r r r

5

Linear acceleration model

Flight under the influence of linear acceleration without gravity attraction.

2

0 12

3 3 2 2

,

,

,

, , : ,

, ,

ij

ij i j

ij i j

CAE ij CAE CAE ij CAE

dt

dt

t t t

t t t

v t r t

ra a a

r r r

v v v

r v r v R R R R

v r

,

0 0, , , ,

0 0 0

arr dep

CAE CAEij ijij ijdep dep arr arr

CAE

t t t

v t vt t t t

r

r v r v

CAEvCAEv CAEv

x

y

CAEr

depr t arrr t

T, , ,

, , , , , , , .arr

dep

t lin CAE CAE dep

lin CAE CAE dep lin CAE CAE dept

v r v t tv r v t t dt a r v t t

t

a a

1

min , , , .2dep

lin CAE CAE dep dep CAEt

v r v t t t t t

6

Result: CAE, , , 0,1... , 212 508.i j CAE CAEna a t n N N

Sort asteroids

6

max max15 10 km, 2km s,r v

26.957km s, 2.246km s - 13 asteroids12

CC

vv

7

Routing problem

8

Routing problem

Asteroids selected at sorting and compiled in a pair are the vertices of a dynamic weighted graph whose

edges carry information about the moment of close approach. Being in one of the nodes of the graph is always

possible to determine the date of the launch, the time and the mass by using a linear acceleration model.

For each flight between two asteroids we consider the following problem:

_ _ _ _ _ _ 11,2... , 10;500 day, , , 30 ,2

iCAE i dep i CAE i dep i dep i dep i arr i

ti N t t t t t t t day

_

_

_

_ _ _

_ _ _ _

_

ln 1 ,

min , , , ,

, , , 0.8 0,

1 .

dep i

problin sp

prob probsp i

ave i prob

i sp dep i

lin CAE i CAE i i dep it

lin CAE i CAE i i dep i ave i

v I

i dep i

I tTa

t I m

a r v t t

a r v t t a

m e m

On this graph we need to find all the routes satisfying the conditions of the main task

_ 0 _ _ 1MJD 59215 MJD 62867, 1200 kg, 30day 6 years.arr i i dep i arr i

i i

t m t t t

To construct the routes used brute force, Dijkstra's algorithm.

Results: 46 933 routes at all:

• 69 with 14 asteroids, 23 of them are cycle routes,

• 46 864 with 13 asteroids, 4 273 of them are cycle routes.

Consequences of this is that the functional value will not exceed:

9

Routing problem

42.J

From the analysis of these routes it was able to create 70 possible solutions to the value of the

primary functional

31.J

10

Trajectory optimization

11


After chaining, the trajectory of each probe is analyzed in terms of the optimal control problem. The purpose

of this approach - maximizing of the second criterion - the functional J ', which is the total mass of the probe at

the time of their return to the mother ship (at the end of the mission):

3

1

' ( )jj f P

j

J m

By applying the decomposition of the initial task, we assume that the final mass of the probe at each flight

from the asteroid to asteroid inside the chain should be maximized. This allows us to use the next functional for

the any phase of flight between the two neighboring asteroids:

1 ( ) minji f PJ m t

We assume that the time points describing the start and end of the trip each probe between neighboring

asteroids chain - are fixed. Then, their variation within a small neighborhood has improved the value of the

functional (1).

To solve the problem using the indirect method of optimization - the Pontryagin maximum principle. As is

well known, the application of the maximum principle reduces the problem of optimal control for the functional

(2) to the boundary.

(1)

(2)

12

The motion of each probe is described by differential equations of motion of a variable mass with low thrust

in a central Newtonian gravitational field.

3

3 3

, , ,

, , ,

prob prob

prob

sp

d d T dm T

dt dt r m dt I

m

r vv r e

r v

Control function:

_ _0 1, 1, , ,dep i arr it t t e

_ _ _ _

_ 1 _ _ 1 _

( ) ( ), ( ) ( ),

( ) ( ), ( ) ( ),

dep i i dep i dep i i dep i

arr i i arr i arr i i arr i

t t t t

t t t t

r r v v

r r v v

_ _ 1

_ 0

( ) ( ),

( ) 2000 .

dep i arr i

dep

m t m t

m t kg

Boundary conditions:


1,2...39.i

13


The Hamilton function of maximum principle :

T T

3

prob prob

v r m prob

sp

T TH

r m I

ψ r e ψ v

The equations for the adjoint system:

Optimality condition:

,

, arg max ,abs H

e

e

optimal control:

П – switching function.

v m

m w

ψ

Transversality condition for the masses:

_( ) 1.m arr it

, , ,x v md H d H d H

dt dt dt m

ψ ψ

x v

0, 0,

, 0,1 , 0,

1, 0,

v

v

ψe

ψ

14


Shown on the previous slide conditions form the boundary value problem, which are unknown in the initial

values for the adjoint variables:

( ) 0,f z

7

_ _ _( ), ( ), ( ) , .T

v dep i r dep i m dep it t t z ψ ψ z

boundary value problem:

Solution of the boundary problem

For the numerical solution of the boundary value problem the following methods and algorithms are used:

1. Continuation method;

2. algorithm HYBRD1 (Burton S. Garbow, Kenneth E. Hillstrom, Jorge J. More) - modification of Powell hybrid

method;

3. BFGS;

4. Evolution Strategy algorithms – CMA-ES.

The above mentioned methods are equally used to solve the basic problem of flight probe between neighboring

asteroids.

15


The resulting trajectory probes were obtained by successive solutions of boundary value problems flights

between asteroids for each of the three chains - in accordance with the selected route. It is worth noting that in the

final analysis of the trajectories of probes times characterizing 'nodes' route varied to increase the final weight of

each unit. As a result, obtained the following results:

MJD undock probes:

63416.52999 (P1), 63619.44824 (P2), 63931.01598 (P3).

Visited asteroids:

P1: 8381, 4307, 6554, 8473, 5657, 2323, 15453, 6295, 1048, 7364;

P2: 1144, 4453, 16038, 12591, 4328, 11081, 14193, 12542, 13986, 16063, 14228;

P3: 11033, 2639, 2100, 961, 15417, 14215, 1265, 1569, 7433, 8093.

MJD docking probes with Mothership :

65608.02999 (P1), 65810.94824 (P2), 66122.01598 (P3).

Final mass probes:

914.12994947735 кг (P1), 876.94771468142 кг (P2), 883.41209585745 кг (P3).

16


Mothership performs the following maneuvers: at the time corresponding 62680.52999 MJD start from

Earth. The magnitude of the hyperbolic excess velocity provides 6 km / s. Coordinates of the first impulse are:

1 5.9406418 0.7814356 0.3132608 kmVs

Further, at the time the relevant 63194.52999 MJD mohership reaches the asteroid in 1144, is a boundary

for the given cyclic route probe. At this point, realized the second impulse rate, providing rendezvous with the

parent asteroid. Components of the second impulse are:

2 -5.3614082 0.6510723 1.1500497 kmVs

The magnitude of the second impulse is: 5.5218844kms

Time of flight of mothership 514 days. Further the mothership don’t maneuver that is moving together with

asteroid 1144.

Trajectory of mothership is the solution of Lambert problem for flight from Earth to the target asteroid.

Date of start and time of flight are chosen with nodal dates of flight path of the probes in order to minimize the

total value of impulse required for the realization of the maneuver.

17


Figure 1. Projection of the Trajectory onto the ecliptic plane

18


Figure 2. Projection of the trajectory onto the XZ plane Figure 3. Projection of the trajectory onto the YZ plane

19

Post-Competition Analysis

20

After the competition our team continue to analyze the problem using the same methodology and was

applied new cycler routes which start from asteroid 1144 and finish at 7364. Realization of such route assume to

additional maneuver of the mothership, flight from first asteroid to the final asteroid. For the considered route

we achieved 32 unique asteroids, so the main functional is equal to 32.

MJD undocking of the probes:

63416.52999 (P1), 63619.4482 (P2), 63.93101598 (P3).

Visited asteroids:

P1: 8381, 4307, 6554, 8473, 5657, 2323, 15453, 6295, 1048, 9710, 7364;

P2: 1144, 4453, 16038, 12591, 4328, 11081, 14193, 12542, 13986, 16063, 14228;

P3: 11033, 2639, 2100, 961, 15417, 14215, 1265, 1569, 7433, 10979, 16030.

MJD docking of the probes with the mothership:

65608.02999 (P1), 65839.40887(P2), 66122.01598 (P3).

Final mass of the probes:

817.498251671007 кг (P1), 806.936738562973 кг (P2), 801.183234822014 кг (P3).


21

Mother ship makes following maneuvers:

First impulse- departure from the Earth - 62680.52999 MJD;

Second impulse– achieving of the first boundary asteroid – 63194.52999 MJD;

Third impulse– departure from the first boundary asteroid – 63986.72000 MJD;

Forth impulse– achieving second boundary asteroid – 64590.50640 MJD.

Final mass of the mother ship with probes- 8900.993087346308 кг.

Final mass of the mother ship without probes - 6038.24212680923 кг.

Main functional J=32.

Second functional J’=2425.61822

Maneuvering of the mother ship:


22

Just considered the problem of end to end trajectory optimization for the probes. As a criterion for the

quality of this task, consider the following functional:

2 ( ) minfJ m t

The functional is analyzed within the time interval:

𝑡0 – departure date from the first asteroid in chain , 𝑡𝑓 – arrival date to the final asteroid in the chain. Thus, we

formulate the problem of maximizing the final mass of the probe during the flight of the entire chain of

asteroids, beginning at the time 𝑡0 until time 𝑡𝑓 . Thus the times approaching asteroids within the chain was not

originally specified, and must be selected (optimized!) for solving the problem.

0 0, , ,f ft t t t fixed

The equations of motion of the probe together with the conditions of the maximum principle, with the

exception of the transversality conditions in the problem of minimizing the functional J2 remain the same and

coincide with the relevant conditions of the problem on the minimum of the functional J1.

terminant:

0

0 0 0 0 0 0 0

0 1

1

( ) ( ) ( ) ( ) ...

( ) ( ) ( ) ( ) ...

( ) ... ( ) ( )

X m

i i i i i i i i

X X

i i i N

t X f f

l m t X t Xa t m t m

X t Xa t X t Xa t

t t t X t Xa t

j - Lagrange multipliers; 6( ) ,T

X t r V X - State vector;

6( ) ,T

i i i i

a aXa t r V Xa - Position and velocity vector i-th asteroid in the chain, i=0,1..11,i it t - Moment of departure and arrival of the probes to the i-th asteroid


23

Transversality conditions:

( ) , ( ) , ( ) ,i i i

i i i

X X t

t t t

l l lt t t

t

ψ ψ

X X

( ) , ( ) , ( )i i i

i i i

t m m

t t t

l l lt t t

t m m

Necessary conditions for optimality:

0it

tH

0it

tH

0 00, 1

Excluding the Lagrange multipliers of the above conditions and complementing their boundary conditions and the

corresponding moment 𝑡0 and 𝑡𝑓, we finally obtain the complete system of necessary optimality conditions:

T T

0

( ) ( )

( ) ( ) ( ) ( ) 0

0

( ) ( ) 0

( )

i i

i i i

X X

i i i i i i i i

Xt t

i i

i i

m m

m

t t

H H t a t a t Xa t

t t t

t t

Tf

ψ ψ ν

ψ X X ν


24

It can be shown that the system of the above conditions, is equivalent to the following system of equations:

.i ii it t

t t

These relations determine the equality of the function switch to the "nodes" of the chain.

the following boundary value problem is formulated:

2

77

2 0 0 2

( ) 0,

( ), ( ), , , 1..10, .i i

X mt t t i

f z

z ψ ν z

Solution of the resulting boundary value problem was achieved by sequentially increasing the number of

asteroids in a chain starting with the one intermediate asteroid; at the same time for each sub-task corresponding

values of 𝑡0 and 𝑡𝑓 were recorded.


25

Results:

The figure shows the switching function as a function of time (dimensionless) during the flight of the probe

under consideration chains asteroids. You can clearly see that the overall structure of the switching control has

changed - there were two fully active segments. The conditions expressed the equal magnitude of switching

function at the nodes.


26

the change of

probe’s mass (kg) as a

function of

dimensionless time

the change of costate

variable to the mass

as a function of

dimensionless time


27

The figure shows the projection of the

trajectory of the probe (P3) on the plane of

the ecliptic, the resulting solution of the

problem end to end optimization. Blue shows

the active phase, the red - passive.

Final mass of the probe obtained in the solution of this problem:

that almost 13 kilograms (kg 801.183234822014 (P3)) higher than the final mass obtained for the third

probe in a phased analysis of flight trajectories in the given solution. The increase in final mass of the

probe is connected both with the optimal timing of approaching the intermediate asteroids, and changes

in program of switching function - in the chain is optimal availability of fully active segment of flight

between the asteroids.

( ) 814.31749 kgfm t


team 36: viacheslav petukhov (riame), roman elnikov (mai

Documents