PROPT Free Floating Robot

This page is part of the PROPT Manual. See PROPT Manual.

Users Guide for dyn.Opt, Example 6a, 6b, 6c

A free floating robot

Problem description

Find u over t in [0; 5 ] to minimize

6c is free end time

6a:

$\int _{0}^{5}0.5*(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2})\mathrm {d} t+$

$(x_{1}(t_{F})-4.0)^{2}+(x_{3}(t_{F})-4.0)^{2}+x_{2}(t_{F})^{2}+x_{4}(t_{F})^{2}+x_{5}(t_{F})^{2}+x_{6}(t_{F})^{2}$

6b:

$\int _{0}^{5}0.5*(u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2})\mathrm {d} t$

6c:

$J=t_{F}$

subject to:

$M=10.0$

$D=5.0$

$Le=5.0$

$In=12.0$

$s5=sin(x_{5})$

$c5=cos(x_{5})$

${\frac {dx_{1}}{dt}}=x_{2}$

${\frac {dx_{2}}{dt}}={\frac {(u_{1}+u_{3})*c5-(u_{2}+u_{4})*s5}{M}}$

${\frac {dx_{3}}{dt}}=x_{4}$

${\frac {dx_{4}}{dt}}={\frac {(u_{1}+u_{3})*s5+(u_{2}+u_{4})*c5}{M}}$

${\frac {dx_{5}}{dt}}=x_{6}$

${\frac {dx_{6}}{dt}}={\frac {(u_{1}+u_{3})*D-(u_{2}+u_{4})*Le}{In}}$

$x(0)=[0\ 0\ 0\ 0\ 0\ 0];$

6b - x(5) = [4 0 4 0 0 0]; 6c - x(5) = [4 0 4 0 pi/4 0]; 6c - -5 <= u <= 5

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t

for i=1:3

    if i==3
        toms t_f
    else
        t_f = 5;
    end
    p1 = tomPhase('p1', t, 0, t_f, 40);
    setPhase(p1);

    tomStates x1 x2 x3 x4 x5 x6
    tomControls u1 u2 u3 u4

    % Initial guess
    if i==1
        x0 = {icollocate({x1 == 0; x2 == 0; x3 == 0
            x4 == 0; x5 == 0; x6 == 0})
            collocate({u1 == 0; u2 == 0
            u3 == 0; u4 == 0})};
    elseif i==2
        x0 = {icollocate({x1 == x1_init; x2 == x2_init
            x3 == x3_init; x4 == x4_init
            x5 == x5_init; x6 == x6_init})
            collocate({u1 == u1_init; u2 == u2_init
            u3 == u3_init; u4 == u4_init})};
    else
        x0 = {t_f == tf_init
            icollocate({x1 == x1_init; x2 == x2_init
            x3 == x3_init; x4 == x4_init
            x5 == x5_init; x6 == x6_init})
            collocate({u1 == u1_init; u2 == u2_init
            u3 == u3_init; u4 == u4_init})};
    end

    % Box constraints
    if i<=2
        cbox = {icollocate({
            -100 <= x1 <= 100; -100 <= x2 <= 100
            -100 <= x3 <= 100; -100 <= x4 <= 100
            -100 <= x5 <= 100; -100 <= x6 <= 100})
            collocate({-1000 <= u1 <= 1000; -1000 <= u2 <= 1000
            -1000 <= u3 <= 1000; -1000 <= u4 <= 1000})};
    else
        cbox = {
            icollocate({-100 <= x1 <= 100; -100 <= x2 <= 100
            -100 <= x3 <= 100; -100 <= x4 <= 100
            -100 <= x5 <= 100; -100 <= x6 <= 100})
            collocate({-5 <= u1 <= 5; -5 <= u2 <= 5
            -5 <= u3 <= 5; -5 <= u4 <= 5})};
    end

    % Boundary constraints
    cbnd = initial({x1 == 0; x2 == 0; x3 == 0
        x4 == 0; x5 == 0; x6 == 0});
    if i==2
        cbnd6b = {cbnd
            final({x1 == 4; x2 == 0
            x3 == 4; x4 == 0
            x5 == 0; x6 == 0})};
    elseif i==3
        cbnd6c = {cbnd
            final({x1 == 4; x2 == 0
            x3 == 4;    x4 == 0
            x5 == pi/4; x6 == 0
            1 <= t_f <= 100})};
    end

    % ODEs and path constraints
    M = 10.0;
    D = 5.0;
    Le = 5.0;
    In = 12.0;
    s5 = sin(x5);
    c5 = cos(x5);

    ceq = collocate({
        dot(x1) == x2
        dot(x2) == ((u1+u3).*c5-(u2+u4).*s5)/M
        dot(x3) == x4
        dot(x4) == ((u1+u3).*s5+(u2+u4).*c5)/M
        dot(x5) == x6
        dot(x6) == ((u1+u3)*D-(u2+u4)*Le)/In});

    % Objective

Solve the problem

    options = struct;
    if i==1
        objective = (final(x1)-4)^2+(final(x3)-4)^2+final(x2)^2+ ...
            final(x4)^2+final(x5)^2+final(x6)^2 + ...
            integrate(0.5*(u1.^2+u2.^2+u3.^2+u4.^2));
        options.name = 'Free Floating Robot 6a';
        solution1 = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
        tp  = subs(collocate(t),solution1);
        x1p = subs(collocate(x1),solution1);
        x2p = subs(collocate(x2),solution1);
        x3p = subs(collocate(x3),solution1);
        x4p = subs(collocate(x4),solution1);
        x5p = subs(collocate(x5),solution1);
        x6p = subs(collocate(x6),solution1);
        u1p = subs(collocate(u1),solution1);
        u2p = subs(collocate(u2),solution1);
        u3p = subs(collocate(u3),solution1);
        u4p = subs(collocate(u4),solution1);
        tf1 = subs(final(t),solution1);
        x1_init = subs(x1,solution1);
        x2_init = subs(x2,solution1);
        x3_init = subs(x3,solution1);
        x4_init = subs(x4,solution1);
        x5_init = subs(x5,solution1);
        x6_init = subs(x6,solution1);
        u1_init = subs(u1,solution1);
        u2_init = subs(u2,solution1);
        u3_init = subs(u3,solution1);
        u4_init = subs(u4,solution1);
    elseif i==2
        objective = integrate(0.5*(u1.^2+u2.^2+u3.^2+u4.^2));
        options.name = 'Free Floating Robot 6b';
        solution2 = ezsolve(objective, {cbox, cbnd6b, ceq}, x0, options);
        x1_init = subs(x1,solution2);
        x2_init = subs(x2,solution2);
        x3_init = subs(x3,solution2);
        x4_init = subs(x4,solution2);
        x5_init = subs(x5,solution2);
        x6_init = subs(x6,solution2);
        u1_init = subs(u1,solution2);
        u2_init = subs(u2,solution2);
        u3_init = subs(u3,solution2);
        u4_init = subs(u4,solution2);
        tf_init = subs(final(t),solution2);
    else
        objective = t_f;
        options.name = 'Free Floating Robot 6c';
        solution3 = ezsolve(objective, {cbox, cbnd6c, ceq}, x0, options);
    end

Problem type appears to be: qpcon
Time for symbolic processing: 0.29507 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Free Floating Robot 6a         f_k      13.016949152618100000
                                       sum(|constr|)      0.000000000121463108
                              f(x_k) + sum(|constr|)     13.016949152739564000
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   35 ConJacEv   35 Iter   31 MinorIter  405
CPU time: 0.374402 sec. Elapsed time: 0.389000 sec. 

Problem type appears to be: qpcon
Time for symbolic processing: 0.26814 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Free Floating Robot 6b         f_k      76.800000142684667000
                                       sum(|constr|)      0.000000241107807079
                              f(x_k) + sum(|constr|)     76.800000383792479000
                                              f(x_0)      6.802639150498418300

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

FuncEv    1 ConstrEv   35 ConJacEv   35 Iter   25 MinorIter  395
CPU time: 0.343202 sec. Elapsed time: 0.336000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.28937 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Free Floating Robot 6c         f_k       4.161676034118863200
                                       sum(|constr|)      0.000000000038961710
                              f(x_k) + sum(|constr|)      4.161676034157824900
                                              f(x_0)      5.000000000000000000

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

FuncEv    1 ConstrEv   26 ConJacEv   26 Iter   16 MinorIter  570
CPU time: 0.218401 sec. Elapsed time: 0.208000 sec. 

end

Plot result

tf2 = tf_init;
tf3 = subs(t_f,solution3);
disp(sprintf('\nFinal time for 6a = %1.4g',tf1));
disp(sprintf('\nFinal time for 6b = %1.4g',tf2));
disp(sprintf('\nFinal time for 6c = %1.4g',tf3));

subplot(2,1,1)
plot(tp,x1p,'*-',tp,x2p,'*-',tp,x3p,'*-',tp,x4p,'*-' ...
    ,tp,x5p,'*-',tp,x6p,'*-');
legend('x1','x2','x3','x4','x5','x6');
title('Free Floating Robot state variables');

subplot(2,1,2)
plot(tp,u1p,'+-',tp,u2p,'+-',tp,u3p,'+-',tp,u4p,'+-');
legend('u1','u2','u3','u4');
title('Free Floating Robot control');

Final time for 6a = 5

Final time for 6b = 5

Final time for 6c = 4.162