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MultipleShooting.m
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MultipleShooting.m
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function [correctedInitialEpoches, correctedInitialStates, exitflag] = MultipleShooting(dynamicFcn, initialEpoches, initialStates, positionTolerance, velocityTolerance, odeOptions)
%multiple shooting method in the given dynamic system
%
% This function will automatically use 'parfor' (after R2017a) if a parpool has been open outside.
%
% References:
% Marchand, Belinda G., Kathleen C. Howell, and Roby S. Wilson. 2007. “Improved Corrections Process for Constrained Trajectory Design in the N-Body Problem.” Journal of Spacecraft and Rockets 44 (4):884–97. https://doi.org/10.2514/1.27205.
% Parker, Jeffrey S., and Rodney L. Anderson. 2014. Low-Energy Lunar Trajectory Design. 1st ed. JPL Deep-Space Communications and Navigation Series, July. Wiley. http://descanso.jpl.nasa.gov/Monograph/series12_chapter.cfm?force_external=0.
%
% see also: PositionShooting
if nargin < 4
positionTolerance = 1e-9;
velocityTolerance = 1e-7;
odeOptions = odeset('AbsTol',1e-12,'RelTol',1e-12);
disp('Default position, velocity, and propagation tolerance is used.');
end
flagAutoUseParallel = 1; % 1: use 'parfor' if find one parallel pool
initialPositionFixed = false; % 为 true 时较容易收敛
finalPositionFixed = false; % 为 true 时较难收敛
iterationNumberLevelTwoMax = 20;
segmentNumber = length(initialEpoches)-1; % the last element is the target epoch, similar to initialStates
iterationNumberLevelTwo = 1;
correctedInitialEpoches = initialEpoches; % will be corrected by multiple shooting method
correctedInitialStates = initialStates; % will be corrected by multiple shooting method
while 1
%---- level-1 shooting ----
% simple shooting to connect position of every segment
stateTransitionMatrixes = zeros(segmentNumber, 6, 6);
exitflag = zeros(segmentNumber,1);
correctedFinalStates = zeros(segmentNumber,6); % used to calculate Delta V
p = gcp('nocreate');
if flagAutoUseParallel && ~isempty( p ) % this will test if a parpool is open, and suppress gcp to creat one
% use parallel
% usually faster, but your dynamicFcn must support parallel computing
disp(['# level-1 shooting begin... (using ' num2str(p.NumWorkers) ' workers)']);
parforCorrectedInitialEpoches = correctedInitialEpoches; % auxiliary variables for parfor, will be discarded after parfor
parforCorrectedInitialStates = correctedInitialStates; % auxiliary variables for parfor, will be discarded after parfor
parfor iiSegment = 1:segmentNumber
[correctedInitialStates(iiSegment,:), correctedFinalStates(iiSegment,:), stateTransitionMatrixes(iiSegment,:,:), exitflagLevelOne(iiSegment)]...
= PositionShooting(...
dynamicFcn,...
correctedInitialEpoches(iiSegment), correctedInitialStates(iiSegment,:),...
parforCorrectedInitialEpoches(iiSegment+1), parforCorrectedInitialStates(iiSegment+1,:),...
positionTolerance, odeOptions);
if exitflagLevelOne(iiSegment) ~= 1
disp(['# !!! fail: segment ' num2str(iiSegment) ' fails at level-1 shooting.']);
else
disp(['# segment ' num2str(iiSegment) ' done.']);
end
end
else
% use serial
% slower, but easier for testing, debugging, and profiling
disp(['# level-1 position shooting begin...']);
for iiSegment = 1:segmentNumber
[correctedInitialStates(iiSegment,:), correctedFinalStates(iiSegment,:), stateTransitionMatrixes(iiSegment,:,:), exitflagLevelOne(iiSegment)]...
= PositionShooting(...
dynamicFcn,...
correctedInitialEpoches(iiSegment), correctedInitialStates(iiSegment,:),...
correctedInitialEpoches(iiSegment+1), correctedInitialStates(iiSegment+1,:),...
positionTolerance, odeOptions);
if exitflagLevelOne(iiSegment) ~= 1
disp(['# !!! fail: segment ' num2str(iiSegment) ' fails at the level-1 shooting.']);
break;
else
disp(['# segment ' num2str(iiSegment) ' done.']);
end
end
end
disp(['# level-1 done.']);
% draw level-1 shooting results
figure(99); clf; PlotInitialState(dynamicFcn, correctedInitialEpoches, correctedInitialStates);
% test failure
if any(exitflagLevelOne ~= 1)
exitflag = -2;
break; % exitflag already set to -2 in above code
end
%---- level-2 shooting ----
% collcect the target error
deltaVelocity = correctedFinalStates(1:segmentNumber-1,4:6) - correctedInitialStates(2:segmentNumber,4:6);
deltaVelocity = reshape( deltaVelocity.', [], 1 );
disp(['# level-2 iter ' num2str(iterationNumberLevelTwo) ' norm: ' num2str(norm(deltaVelocity))]);
% test early stop
if norm(deltaVelocity) < velocityTolerance
exitflag = 1;
disp(['# multiple-shooting success. norm(dV) = ' num2str(norm(deltaVelocity))]);
break;
end
% modify epoch and velocity of all segments at once
% after one modification, shooting position again
stateRelationshipMatrix = zeros( (segmentNumber-2)*3+3, (segmentNumber-2)*4+12 );
for ii = 2:segmentNumber
% generate state relationship matrix
stm21 = squeeze( stateTransitionMatrixes(ii-1, :, :) );
stm12 = inv(stm21);
stm32 = squeeze( stateTransitionMatrixes(ii, :, :) );
%stm23 = inv(stm32); % seems do not need this
%
v1plus = correctedInitialStates(ii-1, 4:6).';
v2minus = correctedFinalStates(ii-1, 4:6).';
v2plus = correctedInitialStates(ii, 4:6).';
v3minus = correctedFinalStates(ii, 4:6).';
%
a2minus = dynamicFcn( correctedInitialEpoches(ii), correctedFinalStates(ii-1, :).' ); a2minus = a2minus(4:6);
a2plus = dynamicFcn( correctedInitialEpoches(ii), correctedInitialStates(ii, :).' ); a2plus = a2plus(4:6);
%
stateRelationshipMatrix( (ii-2)*3+1:(ii-2)*3+3, (ii-2)*4+1:(ii-2)*4+12 ) = [...
-inv(stm12(1:3,4:6)),...
stm12(1:3,4:6)\v1plus,...
-stm32(1:3,4:6)\stm32(1:3,1:3) + stm12(1:3,4:6)\stm12(1:3,1:3),...
(a2plus-a2minus) + (stm32(1:3,4:6)\stm32(1:3,1:3)*v2plus-stm12(1:3,4:6)\stm12(1:3,1:3)*v2minus),...
inv(stm32(1:3,4:6)),...
-stm32(1:3,4:6)\v3minus];
end
%disp(stateRelationshipMatrix);
% solve for the correction
if initialPositionFixed
stateRelationshipMatrix = stateRelationshipMatrix(:, 5:end);
end
if finalPositionFixed
stateRelationshipMatrix = stateRelationshipMatrix(:, 1:end-2);
end
correctionAtPositionAndEpoch = pinv(stateRelationshipMatrix) * deltaVelocity;
% updata initial epoches and states for next iteration
updateSegments = 1:segmentNumber+1;
indexOffset = 0;
if initialPositionFixed
updateSegments = setdiff(updateSegments, 1); % do not update first state if it should be fixed
indexOffset = -1; % to compensate that first element of correctionAtPositionAndEpoch is for the 2nd point
end
if finalPositionFixed
updateSegments = setdiff(updateSegments, segmentNumber+1); % do not update last state if it should be fixed
end
sigma = 1;
for ii = updateSegments
correctedInitialEpoches(ii) = correctedInitialEpoches(ii) + sigma * correctionAtPositionAndEpoch((ii+indexOffset-1)*4+4);
correctedInitialStates(ii,1:3) = correctedInitialStates(ii,1:3) + sigma * correctionAtPositionAndEpoch((ii+indexOffset-1)*4+1:(ii+indexOffset-1)*4+3).';
end
iterationNumberLevelTwo = iterationNumberLevelTwo + 1;
% stop after too many iterations
if iterationNumberLevelTwo > iterationNumberLevelTwoMax
exitflag = -1;
disp(['# !!! fail: level-2 shooting exceeds maximum iteration number ' num2str(iterationNumberLevelTwoMax) '.']);
break;
end
end
% return corrected epoches and states
% correctedInitialEpoches = correctedInitialEpoches;
% correctedInitialStates = correctedInitialStates;
end
function [correctedInitialState, correctedFinalState, correctedStateTransitionMatrix, exitflag] = PositionShooting(dynamicFcn, initialEpoch, initialState, targetEpoch, targetState, positionTolerance, odeOptions)
%[correctedInitialState, correctedFinalState, correctedStateTransitionMatrix, exitflag] = PositionShooting(dynamicFcn, initialEpoch, initialState, targetEpoch, targetState, positionTolerance, odeOptions)
%correctedInitialState = PositionShooting(dynamicFcn, initialEpoch, initialState, targetEpoch, targetState, positionTolerance)
%simple shooting method for position in the given dynamic system
% This is called by MultipleShooting.m for the level-1 shooting
%
% fixed variables: initial epoch, initial position, target epoch, target position
% free variables: initlal velocity
%
% inputs:
% dynamicFcn
% initialEpoch
% initialState
% targetEpoch
% targetState
% positionTolerance
% [odeOptions]: has a default value
% outputs:
% correctedInitialState: shooting result
% correctedFinalState: final state is used by MultipleShooting, but not for most other situations.
% correctedStateTransitionMatrix: required by MultipleShooting to avoid recalculation.
% exitflag: 1 for suceess; -1 for too many iterations. (need to check more cases)
%
% see also: MultipleShooting
%
% References:
% Marchand, Belinda G., Kathleen C. Howell, and Roby S. Wilson. 2007. “Improved Corrections Process for Constrained Trajectory Design in the N-Body Problem.” Journal of Spacecraft and Rockets 44 (4):884–97. https://doi.org/10.2514/1.27205.
% Parker, Jeffrey S., and Rodney L. Anderson. 2014. Low-Energy Lunar Trajectory Design. 1st ed. JPL Deep-Space Communications and Navigation Series, July. Wiley. http://descanso.jpl.nasa.gov/Monograph/series12_chapter.cfm?force_external=0.
% check inputs
if nargin < 7
odeOptions = odeset('AbsTol',1e-9,'RelTol',1e-9);
disp('Default propagation accuracy of 1e-9 is used.');
end
iterationNumberMax = 50;
fixedPropagationInterval = [initialEpoch, (initialEpoch+targetEpoch)/2, targetEpoch]; % insert a middle point to eliminate too much redundant results
iterationNumber = 1;
while 1
% calculate state transition matrix
[~,allState] = ode113( dynamicFcn, fixedPropagationInterval, [initialState,reshape(eye(6),1,36)], odeOptions );
finalState = allState(end, 1:6);
stateTransitionMatrix = reshape( allState(end,7:42).', 6, 6 );
% check if target is reached
errorFinalState = targetState(1:3) - finalState(1:3);
%disp(['debug: position shooting: iter ' num2str(iterationNumber) ': error is ' num2str(norm(errorFinalState(1:3)))]);
% test early stop
if norm(errorFinalState(1:3)) < positionTolerance
exitflag = 1; % success
break;
end
% solve for the correction at initial state
L = stateTransitionMatrix( 1:3, 4:6 );
correctionAtInitialState = ( L \ errorFinalState.' ).';
% update state for next iteration
sigma = 0.618;
initialState(4:6) = initialState(4:6) + sigma * correctionAtInitialState(1:3);
iterationNumber = iterationNumber + 1;
% stop after too many iterations
if iterationNumber > iterationNumberMax
exitflag = -1; % too much iterations
disp(['position shooting: max iteration reached.']);
break;
end
end
% generate output
correctedInitialState = initialState;
correctedFinalState = finalState;
correctedStateTransitionMatrix = stateTransitionMatrix;
end