Team 36: Viacheslav Petukhov (RIAME), Roman Elnikov (MAI), Alexey Ivanyuhin (RIAME), Mikhail Konstantinov (MAI), Min Thein (MAI) , Ilya Nikolichev (MAI), Dien Nguyen Ngoc (MAI) Research Institute of Applied Mechanics and Electrodynamics (RIAME) & Moscow Aviation Institute (MAI) (Russia) 2015
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Team 36: Viacheslav Petukhov (RIAME), Roman Elnikov (MAI),
Alexey Ivanyuhin (RIAME), Mikhail Konstantinov (MAI), Min Thein
817.498251671007 кг (P1), 806.936738562973 кг (P2), 801.183234822014 кг (P3).
Post-Competition Analysis
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:
Post-Competition Analysis
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
Post-Competition Analysis
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
Post-Competition Analysis
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.
Post-Competition Analysis
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.
Post-Competition Analysis
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
Post-Competition Analysis
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