Madrid New Optimal Strategies for the Station Keeping of - - PowerPoint PPT Presentation
UCM Modelling Week 16 th -24 th June 2008 Madrid New Optimal Strategies for the Station Keeping of Communications Satellites in Geostationary Orbits using Electric Propulsion Problem proposed by GMV Miguel ngel Henche, Matthew Edwards, Jos
Problem proposed by GMV
1.
Statement of the problem: optimal control for a dynamical system
2.
Model for the dynamical system: second order differential equations
3.
Solution with the method of variation of the constants
4.
Analysis of the evolution of parameters involved
5.
Control linear equations
6.
Optimization control minimizing a cost function
7.
Assumptions for determining the cost function
8.
Definition of the cost function
9.
Algorithm for minimize the cost function
10.
Analysis of the results
11.
Future works
Earth Gravitational Field Lunisolar Force Solar Radiation Pressure
Orbital station keeping manoeuvres for a geostationary satellite are performed to compensate for natural perturbations that tends to change the orbit to non geostationary. Station keeping Modelling: Mean orbital elements: obtained by means of linearized Lagrange equations, where the perturbation function contains only those terms causing secular and long period perturbations. Linear equations for computing manoeuvres Classical Approach Two thrusters located in normal plane (N/S) and in tangential plane(E/W) New Model (proposed by GMV): One thruster with direction specified by the cant, γ, and, σ, slew angles.
Problem definition Objective function Equality constraints Inequality constraints Objective function Optimisation variables for each manoeuvre: Mid-point of the manoeuvre Duration of the manoeuvre
m m i x g m i x g x x f
e i e i n
,..., ) ( ,..., 1 ) ( : ) ( min
1 +
= ≥ = = ℜ ∈
= n manoeuvre manoeuvre
Mass
1
min
Geostationary satellites have e and i values close to
ex = e cos(Ω + ω) ey = e sin(Ω + ω)
ix = i cosΩ iy = i sinΩ
Lagrange equations
The inclination vector is modified:
0º.3895cos 0º.00457cos2( ) 0º.02331cos2 0º.8475 0º.2903sin 0º.004sin 2( ) 0º.02139sin 2
x L L L y L L L
i i t
Periodical
ω υ λ ω υ λ = − Ω − + − = − Ω − + −
( ) ( ) (cos ( ) cos ( )), ( ) ( ) (sin ( ) sin ( )),
x x e y y e
e t e t R s t s t e t e t R s t s t = + − = + −
The evolution of the mean longitude is parabolic The evolution of the mean inclination vector has
The annual evolution of the mean eccentricity
2 cos( ) sin( ) 2 2 sin( ) cos( ) cos( ) sin( )
t r x b b t t r y b b n n x b r n y b
V V e s s V V e V V V V e s s V V V V i V i s V V e V V i s V ⎫ Δ = + ⎪ ⎧ Δ ⎪ ⎪Δ = ⎪ ⎪ Δ = − ⎪ ⎪ ⇒ Δ = Δ ⎬ ⎨ ⎪ ⎪ Δ = − Δ = Δ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ Δ = − ⎭
Fix the longitude ls=30ºW Fix the longitude and latitude dead-bands to be
Fix the year to be 2008. So Δi=0.9173,
Consider each day separately and assume that
2
Assume only 1 thruster, whose direction is defined by a cant angle, γ, and slew angle, σ. Assume the thruster is a Stationary Plasma Thruster, which gives F=61.5 X 10-3 N. Assuming the mass of the satellite is 4000kg we get an acceleration
2 3
t n r d b d
1 2 t r
2 3 t n
2 2 4 4 4 4
b x b y
2
c
Where
sec. 4
365
year
i c Δ =
1 2 3
e
−
+ =
+ −
b b
+
Daily thrust running time
1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 01/01/2008 01/02/2008 01/03/2008 01/04/2008 01/05/2008 01/06/2008 01/07/2008 01/08/2008 01/09/2008 01/10/2008 01/11/2008 01/12/2008
Days Time duration of thrust
Td
5 10 15 20 25 Hour 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65
Number of Days
Sidereal time for the mid-point of the manouver
Sb