Contents
- Demonstrate 3D optimization.
- Constants
- Mission set up
- Sail physical parameters
- Cost function weighting
- ode113 parameters
- Optimization parameters
- Simplex parameters
- GA parameters
- Costate to sail angle function
- On the next pass use the newly computed lambda as the first guess
- Acceleration
- Data for selected mission
- Dynamics function
- Cost function called by optimizer
- Initial costate guess
- Integration (ode113) parameters
- If nondimensionalize
- Weights for the costates
- Optimize the trajectory.
Demonstrate 3D optimization.
You can select an Earth to Mars mission or a Solar Polar Imager mission. You can pick either dimensional units or nondimensional units. Try either Simplex or genetic algorithms.
Functions demonstrated:
TrajectoryOptimization LambdaToConeClock3D RHSOpt3DOrbit Plot3DTrajectory Cost3D
------------------------------------------------------------------------ See also Constant, IsVersionAfter, Planets, ElToMEq, TimeDisplay, TrajectoryOptimization, LambdaToConeClock3D, Plot3DTrajectory, RHSOpt3DOrbit, Cost3D ------------------------------------------------------------------------
%------------------------------------------------------------------------------- % Copyright (c) 2005-2006 Princeton Satellite Systems, Inc. % All rights reserved. %------------------------------------------------------------------------------- % Since version 7. %------------------------------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%% % User parameters %%%%%%%%%%%%%%%%%%%%% mission = 'elements'; % 'elements' 'planet name', 'spi', 'inclination' nonDim = 0; % = 1 for nondimensional parameters, mu = 1, normalize position to 1 au method = 'downhill simplex'; % 'downhill simplex' or 'genetic algorithm' % Also see below for many more parameters. %%%%%%%%%%%%%%%%%%%%%
Constants
%---------- aU = Constant('au'); c = Constant('speed of light'); muSun = Constant('mu sun');
Mission set up
%--------------- % general purpose setup for 'elements' mission: el0 = [ aU;0;0;0;0.0;0]; elF = [1.5*aU;0;0;0;0.0;0];
Sail physical parameters
%------------------------- sailArea = 1e4; sailMass = 100; sailRhoS = 1.0; sailRhoAD = 0.0; sailRhoR = 0.0; duration = 420; % days, 420 good for pure planar and 10 deg inclination change
Cost function weighting
%------------------------ % [1;1e8;1e8;4e8;4e8;0.9e9];, Best results planar with simplex % [10;2e9;2e9;1e8;1e8;0.9e9];, Best results inclination with simplex % This weighting is for the 1 aU to 1.5 aU circular orbit transfer problem % No e error, i < 0.1 deg, no a error optimizationErrorWeighting = [1;1e8;1e8;4e8;4e8;0.9e9];
ode113 parameters
%------------------
integrationRelativeTolerance = 1e-2;
integrationAbsoluteTolerance = 1e-2;
Optimization parameters
%------------------------
d.method = method;
d.repeat = 0;
Simplex parameters
%-------------------
d.d.stateTol = 1e-2;
d.d.funTol = 1e-2;
d.d.nIts = 2000;
GA parameters
%-------------- d.d.nG = 50; % generations d.d.nI = 10; % Note: need to scale # mutations and # gen with # individuals. d.d.bounds = [-1 1; -1 1; -1 1; -1 1; -1 1; -1 1]; if IsVersionAfter(6.0) eval('fncSelect = @normGeomSelect;fncXOver = @heuristicXover;fncMutate = @nonUnifMutation;'); else fncSelect = 'normGeomSelect'; fncMutate = 'nonUnifMutation'; fncXOver = 'heuristicXover'; end d.d.fncMutate = fncMutate; d.d.optMutate = [floor(d.d.nI/2) d.d.nG 3]; % # mutations, last gen, sigma d.d.fncSelect = fncSelect; d.d.optSelect = [0.01]; % %chance that a solution is the best d.d.fncXOver = fncXOver; d.d.optXOver = [floor(d.d.nI/2) 3]; % # crossovers, # retries switch method case 'downhill simplex' nSteps = d.d.nIts; case 'genetic algorithm' nSteps = d.d.nG*(2*sum(d.d.optXOver)+d.d.optMutate(1)) + d.d.nI; end
Costate to sail angle function
%------------------------------- if IsVersionAfter(6.0) eval('d.d.lambdaToAccel = @LambdaToConeClock3D;'); else d.d.lambdaToAccel = 'LambdaToConeClock3D'; end
On the next pass use the newly computed lambda as the first guess
%------------------------------------------------------------------
d.d.mu = muSun;
Acceleration
%------------- d.d.accel = sailArea*1367*aU^2*1e-6/(c*sailMass); % km/sec^2
Data for selected mission
%-------------------------- switch mission case 'elements' d.d.xF = ElToMEq( elF, d.d.mu ); d.d.x0 = ElToMEq( el0, d.d.mu ); a0 = el0(1); d.d.iF = elF(2); case 'spi' [name,a,e,i,W,w,L] = Planets('rad',3); % Earth a = a*aU; d.d.x0 = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); a0 = a(1,1); a(1,1) = 0.48*aU; i(1,1) = 70*pi/180; d.d.xF = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); d.d.iF = i(1,1); % Final inclination case 'spimini' [name,a,e,i,W,w,L] = Planets('rad',3); % Earth a = a*aU; d.d.x0 = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); a0 = a(1,1); a(1,1) = 0.9*aU; i(1,1) = 7*pi/180; d.d.xF = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); d.d.iF = i(1,1); % Final inclination otherwise [~,a,e,i,W,w,L] = Planets('rad',3); % Earth a = a*aU; d.d.x0 = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); [name,a,e,i,W,w,L] = Planets('rad',mission); a = a*aU; a0 = a(1,1); d.d.xF = ElToMEq( [a(1,1);i(1,1);W(1,1);w(1,1);e(1,1);L(1,1)], d.d.mu ); d.d.iF = i(1,1); % Final inclination end d.d.xF(6) = 0; % Lambda for state 6 must be zero
Dynamics function
%------------------ d.d.rhsFun = 'RHSOpt3DOrbit'; d.d.plotFun = 'Plot3DTrajectory'; d.d.coordType = 'equinoctial'; % 'cartesian' d.d.boundCond = [1:5 12]; d.d.hasThruster = false; d.d.rhoS = sailRhoS; d.d.rhoAD = sailRhoAD; d.d.rhoR = sailRhoR; TimeDisplay('initialize','Sail 3D Optimization', nSteps );
Cost function called by optimizer
%---------------------------------- if IsVersionAfter(6.0) eval('d.d.costFun = @Cost3D;'); else d.d.costFun = 'Cost3D'; end
Initial costate guess
%----------------------
d.d.lambda0 = [0.0;0;0;0.0025;0.0025;0.00];
Integration (ode113) parameters
%-------------------------------- d.d.data.absTol = integrationAbsoluteTolerance; d.d.data.relTol = integrationRelativeTolerance; d.d.data.maxStep = 1; % ode113 will never take a step bigger than this % fraction of the end time d.d.tEnd = duration*86400; % (s)
If nondimensionalize
%--------------------- d.d.nonDim = nonDim; if( nonDim ) w0 = sqrt(d.d.mu/a0^3); d.d.accel = d.d.accel/(w0^2*a0)/aU^2; d.d.tEnd = d.d.tEnd*w0; d.d.x0(1,1) = d.d.x0(1,1)/a0; d.d.xF(1,1) = d.d.xF(1,1)/a0; d.d.mu = 1; d.d.w0 = w0; d.d.a0 = a0; end
Weights for the costates
%-------------------------
d.d.errorScale = optimizationErrorWeighting;
Optimize the trajectory.
It will use Plot3DTrajectory to plot the results
%------------------------------------------------------- TrajectoryOptimization( d ); TimeDisplay('close' ); %--------------------------------------
Case 1: Method: downhill simplex
Iteration Func-count min f(x) Procedure
0 1 1.24557e+08
1 7 1.21068e+08 initial simplex
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Warning: Failure at t=3.628224e+07. Unable to meet integration tolerances
without reducing the step size below the smallest value allowed (1.192093e-07)
at time t.
31 64 3.90056e+07 reflect
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Warning: Failure at t=3.593226e+07. Unable to meet integration tolerances
without reducing the step size below the smallest value allowed (1.192093e-07)
at time t.
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Warning: Failure at t=3.558291e+07. Unable to meet integration tolerances
without reducing the step size below the smallest value allowed (1.192093e-07)
at time t.
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Exiting: Maximum number of function evaluations has been exceeded
- increase MaxFunEvals option.
Current function value: 3615717.104302
