Max Radius Orbit Transfer

Contents

Problem description

Maximum radius orbit transfer of a spacecraft.

Applied Optimal Control, Bryson & Ho, 1975. Example on pages 66-69.

Programmers: Gerard Van Willigenburg (Wageningen University) Willem De Koning (retired from Delft University of Technology)

% Copyright (c) 2009-2009 by Tomlab Optimization Inc.

Problem setup

% Array with consecutive number of collocation points
narr = [20 40];

toms t;
tf = 1; % Fixed final time

for n=narr

    p = tomPhase('p', t, 0, tf, n);
    setPhase(p)

    tomStates x1 x2 x3
    tomControls u1

    % Parameters
    r0 = 1; mu = 11; th = 1.55;
    m0 = 1; rm0 = -0.25;

    % Initial state
    xi=[r0; 0; sqrt(mu/r0)];

    % Initial guess
    if n==narr(1)
        x0 = {icollocate({x1 == xi(1); x2 == xi(2); x3 == xi(3)})
            collocate({u1 == 0})};
    else
        x0 = {icollocate({x1 == xopt1; x2 == xopt2; x3 == xopt3})
            collocate({u1 == uopt1})};
    end

    % Boundary constraints
    cbnd = {initial({x1 == xi(1); x2 == xi(2); x3 == xi(3)})
        final({x3 == sqrt(mu/x1); x2 == 0})};

    % ODEs and path constraints
    dx1 = x2;
    dx2 = x3.*x3./x1-mu./(x1.*x1)+th*sin(u1)./(m0+rm0*t);
    dx3 = -x2.*x3./x1+th*cos(u1)./(m0+rm0*t);

    ceq = collocate({
        dot(x1) == dx1
        dot(x2) == dx2
        dot(x3) == dx3});

    % Objective
    objective = -final(x1);

Solve the problem

    options = struct;
    options.name = 'Spacecraft';
    solution = ezsolve(objective, {cbnd, ceq}, x0, options);

    xopt1 = subs(x1,solution);
    xopt2 = subs(x2,solution);
    xopt3 = subs(x3,solution);
    uopt1 = subs(u1,solution);

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Spacecraft                     f_k      -1.526286384516587700
                                       sum(|constr|)      0.000000347889260101
                              f(x_k) + sum(|constr|)     -1.526286036627327600
                                              f(x_0)     -0.999999999999998220

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   37 ConJacEv   37 Iter   20 MinorIter   69
CPU time: 0.125000 sec. Elapsed time: 0.125000 sec. 

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Spacecraft                     f_k      -1.526020723222379500
                                       sum(|constr|)      0.000000732944259713
                              f(x_k) + sum(|constr|)     -1.526019990278119700
                                              f(x_0)     -1.526286384516577500

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv  132 ConJacEv  132 Iter   47 MinorIter  209
CPU time: 0.781250 sec. Elapsed time: 0.797000 sec. 

end

% Get final solution
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
u1 = subs(collocate(u1),solution);

%Bound u1 to [0,2pi]
u1 = rem(u1,2*pi); u1 = (u1<0)*2*pi+u1;

% Plot final solution
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Spacecraft states');

subplot(2,1,2)
plot(t,u1,'+-');
legend('u1');
title('Spacecraft controls');