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Investigation of Low Thrust Optimal Orbital Transfer from LEO to GEO Considering Circular Orbits

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Abstract

In the present research, optimal control problem for low thrust spacecraft orbital transfer is investigated by combination of analytical method and Artificial Bee Colony (ABC) algorithm, which is a population-based method. Spacecraft trajectory optimization is defined for orbital transfer from Low Earth Orbit (LEO) to Geosynchronous Earth Orbit (GEO). In the proposed analytical method co-state variables are simplified by employing concepts of optimal control and specific conditions which are considered for the problem, consequently, an adjoint differential equation is added to set of differential equations and these equations are solved simultaneously that result in optimum transfer trajectory. Dynamic modeling of the problem is represented based on Edelbaum equations and trajectory design is carried out considering circular orbits. Optimization problem is defined for different kinds of objective functions such as minimum time, minimum control effort, minimum time with minimum control effort as well as both minimum time and minimum control effort considering different optimal weighted coefficients. These coefficients are searched by means of artificial bee colony algorithm in weighted minimum time minimum effort scheme. Initial values of control variable and terminal time for all defined objective functions are foraged through artificial bee colony algorithm. The rate of changes of the states and control variables are obtained and depicted during orbit transfer. The obtained results demonstrate considerable convergence as well as sufficient accuracy.

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Abbreviations

O:

Orbital velocity

T :

Thrust magnitude

a :

Semi major axes

i :

Orbital inclination

e :

Eccentricity

ω :

Argument of periapsis

Ω :

Longitude of ascending node

θ :

True anomaly

β :

Out of plane control angle of thruster

a 0 :

Initial semi major axes

a f :

Final semi major axes

m :

Space craft mass

\( \overline{x} \) :

State variables

μ :

Earth Gravity constant

H :

Hamiltonian

J :

Cost function

λ i :

Inclination Co-state variable

λ v :

Velocity Co-state variable

F i :

Penalty function

f i :

Objective function

γ :

Ratio of thrust to mass

v ij :

New food position vector in artificial bee colony

G ij :

Food position vector in artificial bee colony

Q ij :

Random number in artificial bee colony

G sj :

Random food position source

w i :

Penalty function weight

k :

Objective function weight

P i :

Probability value

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Correspondence to Mahdi Fakoor.

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Appendix

Appendix

In order to compare and prove the necessity of employing ABC algorithm, in the following the analytical solution for different cases such as (k1=1, k2=0), (k1=0, k2=1) and (k1=1, k2=1) are derived.

Case (I):k1=1,k2=0 we will have MTJ =  ∫ dtas objective function

The solving process will be followed up the same as presented procedure in the manuscript (Eqs. 520) for k1=1, k2=0. In this case using \( \frac{d\beta}{d t}=\frac{\gamma Sin\beta}{V} \) (Eq. 20) as well as Eq. 3, the new set of equations are achieved as follows.

$$ \left\{\begin{array}{c}\frac{d i}{d\beta}=\frac{2}{\pi}\\ {}\frac{d V}{d\beta}=-V\frac{Cos\left(\beta \right)}{Sin\left(\beta \right)}\end{array}\right. $$
(25)

In this method, equations are represented with respect to the β. Also, by solving the above set of equations and putting the final values of i and β, state variables can be attained as follows:

$$ \left\{\begin{array}{c}i={i}_f+\frac{2}{\pi}\left(\beta -{\beta}_f\right)\\ {}V={V}_f\frac{Sin_{\beta_f}}{Sin_{\beta }}\end{array}\right. $$
(26)

Therefore, the relations between initial and final values of i and β are shown as follows:

$$ {i}_f-{i}_0=\frac{2}{\pi}\left({\beta}_0-{\beta}_f\right) $$
(27)
$$ {V}_f\ Sin\ {\beta}_f={V}_0\ Sin\ {\beta}_0 $$
(28)

Subsequently, the time of low-thrust transfer is obtained considering Eq. 25 and Eq. 20 for k1=1, k2 =0:

$$ {T}_f={\int}_0^{t_f} d t={\int}_{\beta_0}^{\beta_f}\frac{V_f\frac{Sin_{\beta_f}}{Sin_{\beta }}}{\gamma Sin\beta} d\beta $$
(29)

As an verification example, for MT objective function in Table 2, by putting the initial value of control variable β0 in Eq. (25) the correct final value for (βf) equal to (−1.78 rad) is obtained that results in the reported control effort (∆β) equal to 1.369 rad. Furthermore from Eq. (29) terminal transfer time is 245.4 (days) which is approximately equal to the reported transfer time in Table 2 (i.e. and 248.8(days)).

Case (II): k 1 =0 and k 2 =1, we will have ME \( \boldsymbol{J}={\int}_{\mathbf{0}}^{{\boldsymbol{t}}_{\boldsymbol{f}}}{\boldsymbol{\beta}}^{\mathbf{2}}\boldsymbol{dt} \) as objective function

Similarly, based on represented equations in the manuscript (Eqs. 520), letting k1=0, k2=1 and given that \( \frac{d\beta}{d t}=\frac{\upbeta \left(\sin \Big(\beta \right)\beta +2\cos \left(\beta \right)\Big)\gamma }{V\left(2+{\beta}^2\right)} \) (Eq. 20) as well as Eq. 3, the new set of equations are achieved as follows:

$$ {\displaystyle \begin{array}{c}\frac{d i}{d\beta}=\frac{2\left(2+{\beta}^2\right) Sin\left(\beta \right)}{\pi \beta \left(\beta Sin\left(\beta \right)+2 Cos\left(\beta \right)\right)}\\ {}\frac{d V}{d\beta}=-\frac{V\left(2+{\beta}^2\right) Cos\left(\beta \right)}{\beta^2 Sin\left(\beta \right)+2\beta Cos\left(\beta \right)}\end{array}} $$
(30)

Likewise, knowing final values of i and β, state variables can be obtained as follows:

$$ {\displaystyle \begin{array}{l}i={i}_f-{\int}_{\beta_0}^{\beta_f}\frac{\beta^2 Si n\left(\beta \right)+2\beta Cos\beta}{V\left(2+{\beta}^2\right)} d\beta \\ {}V={V}_f\frac{\left({\beta_f}^2 Si{n}_{\beta_f}+2{\beta_f}^2 Co{s}_{\beta_f}\right)}{\beta^2 Si{n}_{\beta }+2{\beta}^2 Co s\beta}\end{array}} $$
(31)

Subsequently, the time of orbital transfer is obtained considering Eq. 31 and Eq. 20 for k1=0, k2 =1:

$$ {T}_f={\int}_0^{t_f} d t={\int}_{\beta_0}^{\beta_f}\frac{V_f\frac{\left({\beta_f}^2{Sin}_{\beta_f}+2{\beta_f}^2{Cos}_{\beta_f}\right)\left(2+{\beta}^2\right)}{\beta^2{Sin}_{\beta }+2{\beta}^2 Cos\beta}}{\upbeta \left(\sin \Big(\beta \right)\beta +2\cos \left(\beta \right)\Big)\gamma } d\beta $$
(32)

Case (III):k1= k2=1 we will have MT-MEJ =  ∫ 1+ β2dtas objective function

Similarly, letting k1=1, k2=1 and given that \( \frac{d\beta}{d t}=\frac{\left(\sin \left(\beta \right){\beta}^2\right)+\sin \left(\beta \right)+2\cos \left(\beta \right)\beta \Big)\gamma }{V\left(3+{\beta}^2\right)} \) from (Eq. 20) as well as Eq. 3, the new set of equations are obtained as follows:

$$ \left\{\begin{array}{c}\frac{d i}{d\beta}=\frac{2\left(3+{\beta}^2\right) Sin\left(\beta \right)}{\pi \left({\beta}^2 Sin\left(\beta \right)+2\beta Cos\left(\beta \right)+ Sin\left(\beta \right)\right)}\\ {}\frac{d V}{d\beta}=-\frac{V\left(3+{\beta}^2\right) Cos\left(\beta \right)}{\beta^2 Sin\left(\beta \right)+2\beta Cos\left(\beta \right)+ Sin\left(\beta \right)}\end{array}\right. $$
(33)

Solving the above set of equations and putting the final values of i and β, state variables can be obtained as follows:

$$ {\displaystyle \begin{array}{l}i={i}_f-{\int}_{\beta_0}^{\beta_f}\frac{2\left(3+{\beta}^2\right) Sin\left(\beta \right)}{\pi {\beta}^2 Sin\left(\beta \right)+2\beta Cos\left(\beta \right)+ Sin\left(\beta \right)} d\beta \\ {}V={V}_f\frac{\left({\beta_f}^3 Si{n}_{\beta_f}+2{\beta_f}^2 Co{s}_{\beta_f}+{\beta}_f Sin{\beta}_f\right)}{\beta^3 Si{n}_{\beta }+2{\beta}^2 Co s\beta +\beta Sin\beta}\end{array}} $$
(34)

Therefore, the time of orbital transfer is obtained considering Eq. 34 and Eq. 20 for k1=1, k2 =1 as follows:

$$ {T}_f={\int}_0^{t_f} d t={\int}_{\beta_0}^{\beta_f}\frac{V_f\frac{\left({\beta_f}^3{Sin}_{\beta_f}+2{\beta_f}^2{Cos}_{\beta_f}+{\beta}_f Sin{\beta}_f\right)}{\beta^3{Sin}_{\beta }+2{\beta}^2 Cos\beta +\beta Sin\beta}}{\left(\sin \left(\beta \right){\beta}^2\right)+\sin \left(\beta \right)+2\cos \left(\beta \right)\beta \Big)\gamma } d\beta $$
(35)

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Fakoor, M., Sadeghi, S. & Bakhtiari, M. Investigation of Low Thrust Optimal Orbital Transfer from LEO to GEO Considering Circular Orbits. J Astronaut Sci 67, 77–97 (2020). https://doi.org/10.1007/s40295-019-00184-1

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