Spacecraft trajectory optimization with GA in Matlab (on/off constant thrust)

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Hello.
I would like to kindly ask for support or any advice on how to implement my problem in Matlab, perhaps using the (Global) Optimization Toolbox, and whether it is even possible.
My goal is to find a time-history of control (σ,) during a fuel-optimal spacecraft rendezous with constant low-thrust.
Problem description:
The control variables are defined as:
  • σ - total thrust acceleration
  • - thrust acceleration projections
And the state vector is:
The objective is to minimise:
subject to:
  • The state equations in state-space representation (CW equations):
  • Control variables constraint: and σ can be equal to either 0 or (on/off)
  • Initial conditions: given
  • Terminal constraints: given
  • Final time inequality constraint:
As I understand, this problem can be categorised as a dynamical optimization problem, that involves integer programming. Could it be solved in Matlab, perhaps using the Genetic Algorithms which I believe allow for integer programming?
Are there any available examples on how to implement a spacecraft (or not necessarily spacecraft) trajectory optimization problem in Matlab, using GA? I have been looking for examples for a very long time, but I could not find any. In fact, I could not find any examples even without the on/off thursting constraints, and I would be grateful if anyone could direct me to other spacecraft trajectory optimization implementations in Matlab, perhaps using the fmincon function.
Thank you very much.
  4 Comments
Sam Chak
Sam Chak on 25 Aug 2022
I'm just testing on the dynamics, and I want to see what objective function would I choose if I want to optimize the trajectory in terms of fastest arrival time, minimum error, minimum effort, subject to the constraint:
[t, x] = ode45(@system, [0 10], [0.9; 0.6; 0.3; 0; 0; 0]);
plot(t, x(:,1:3), 'linewidth', 1.5)
grid on, xlabel('t'), ylabel('y(t)'), % ylim([-0.2 1.2])
function dxdt = system(t, x)
dxdt = zeros(6, 1);
% parameters
xf = 0.6; % final x-position
yf = 0.3; % final y-position
zf = 0.9; % final z-position
n = 1;
sigma = 1;
ux = - 2*x(4) - (x(1) - xf) - (2*n*x(5) + 3*(n^2)*x(1));
uy = - 2*x(5) - (x(2) - yf) - (-2*n*x(4));
uz = - 2*x(6) - (x(3) - zf) - (-(n^2)*x(3));
% the dynamics
dxdt(1) = x(4);
dxdt(2) = x(5);
dxdt(3) = x(6);
dxdt(4) = 2*n*x(5) + 3*(n^2)*x(1) + sigma*ux;
dxdt(5) = -2*n*x(4) + sigma*uy;
dxdt(6) = -(n^2)*x(3) + sigma*uz;
end

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Answers (1)

Alan Weiss
Alan Weiss on 21 Aug 2022
You might be interested in this example: Discretized Optimal Trajectory, Problem-Based. The problem formulation is different than yours, so it is probably not directly applicable, but you might be able to make it work for you. One thing to note: I recently found out that this sort of optimal trajectory problem works better when you lower the optimality tolerance, as described here: https://www.mathworks.com/matlabcentral/answers/1774135-possible-bug-with-coneprog.
Alan Weiss
MATLAB mathematical toolbox documentation
  10 Comments
Yakov Bobrov
Yakov Bobrov on 25 Aug 2022
Hi Sam, in the Hohmann transfer, two impulsive maneuvers are performed, while I am trying to model continuous thrust. There is literature available where this was done, but I am tyring to understand how to implement it.

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