Demonstrate the 2D trajectory optimization function for low thrust.
This demo is the same as the built-in demo in TrajectoryOptimization, a low-thrust transfer to Mars. This is a well-studied problem. The method used is downhill simplex.
Things to try: see how the methods do with different initial costates.
See also Constant, Planets, TrajectoryOptimization
Contents
%-------------------------------------------------------------------------- % Copyright (c) 2005 Princeton Satellite Systems, Inc. % All rights reserved. %-------------------------------------------------------------------------- % Since version 7. %-------------------------------------------------------------------------- disp('This demo takes several minutes to run.') d = struct; d.d = CostLowThrust2D( 'get' );
This demo takes several minutes to run.
1. Use downhill simplex (fminsearch)
%--------------------------------- tic d.method = 'downhill simplex'; % On the next pass use the newly computed lambda as the first guess %------------------------------------------------------------------ d.repeat = 1; % Spacecraft properties %---------------------- m0 = 10000*Constant('lb force to kg'); % kg d.d.thrust = 0.0005; % N d.d.mDot = 6.7727e-5; % thrust/uE % Target planet %-------------- [name,a] = Planets('rad',4); % Mars % Heliocentric system %--------------------- d.d.mu = Constant('mu sun'); aU = Constant('au'); aF = a(1)*aU; d.d.rF = aF; % Dynamics function %------------------ d.d.rhsFun = @FLowThrust2D; d.d.plotFun = @Plot2DTrajectory; % Simplex parameters %------------------- d.d.stateTol = 1e-6; d.d.funTol = 1e-6; d.d.nIts = 2500; % Cost function called by simplex %-------------------------------- d.d.costFun = @CostLowThrust2D; % Getting messages about integration toleranges which do not affect solver warning off MATLAB:ode113:IntegrationTolNotMet; % Initial costate guess %---------------------- d.d.lambda0 = [0;-1;1]*1e5; Plot2DTrajectory(d.d.lambda0,d.d) % [r;rDot;v;m] %--------------------------------------------- v0 = VOrbit(aU,aU,d.d.mu); vF = VOrbit(aF,aF,d.d.mu); d.d.x0 = [aU;0;v0;m0]; d.d.xF = [d.d.rF;0;vF]; % Integration (ode113) parameters %-------------------------------- d.d.data.absTol = 1e-10; d.d.data.relTol = 2.5e-8; d.d.data.maxStep = 0.1; % ode113 will never take a step bigger than this % fraction of the end time d.d.tEnd = 1000*86400; % (s) % Weights for the costates and error in the mass equation %-------------------------------------------------------- d.d.errorScale = [1; 3e6; 1e7; 1e-2]; % Optimize the trajectory. It will use Plot2DTrajectory to plot the results %------------------------------------------------------- TrajectoryOptimization( d ); toc %-------------------------------------- % PSS internal file version information %--------------------------------------
Radial position error = 57463397.1744 (km)
Radial velocity error = -0.2684 (km/s)
Tangential velocity error = 1.7734 (km/s)
Time of minimum error = 7.5254 month
ans =
1.7047e+08
-0.26837
25.903
Case 1: Method: downhill simplex
Iteration Func-count min f(x) Procedure
0 1 6.01431e+07
1 4 5.95565e+07 initial simplex
2 6 5.80463e+07 expand
3 7 5.80463e+07 reflect
4 9 5.61238e+07 expand
5 11 5.29135e+07 expand
6 12 5.29135e+07 reflect
7 14 4.71632e+07 expand
8 16 4.18598e+07 expand
9 17 4.18598e+07 reflect
10 19 3.56763e+07 expand
11 21 3.32293e+07 expand
12 22 3.32293e+07 reflect
13 24 3.18217e+07 expand
14 26 3.12576e+07 expand
15 27 3.12576e+07 reflect
16 29 3.10677e+07 expand
17 31 3.08714e+07 expand
18 32 3.08714e+07 reflect
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20 35 3.07756e+07 expand
21 36 3.07756e+07 reflect
22 38 3.06753e+07 expand
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25 42 3.05868e+07 expand
26 44 3.04588e+07 expand
27 45 3.04588e+07 reflect
28 47 3.03918e+07 expand
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100 174 1.7459e+06 reflect
101 176 1.7459e+06 contract outside
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Optimization terminated:
the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-06
and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-06
Radial position error = 2187.3965 (km)
Radial velocity error = -0.0036 (km/s)
Tangential velocity error = -0.0010 (km/s)
Time of minimum error = 1.3053 year
Elapsed time is 13.078521 seconds.
