Free Floating Robot

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

A free floating robot

Contents

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 tf
    else
        tf = 5;
    end
    p1 = tomPhase('p1', t, 0, tf, 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 = {tf == 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 <= tf <= 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 = tf;
        options.name = 'Free Floating Robot 6c';
        solution3 = ezsolve(objective, {cbox, cbnd6c, ceq}, x0, options);
    end

Problem type appears to be: qpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Free Floating Robot 6a         f_k      13.016949152618103000
                                       sum(|constr|)      0.000000000121647941
                              f(x_k) + sum(|constr|)     13.016949152739752000
                                              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  328
CPU time: 0.906250 sec. Elapsed time: 0.906000 sec. 

Problem type appears to be: qpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Free Floating Robot 6b         f_k      76.800001530400650000
                                       sum(|constr|)      0.000000005285607311
                              f(x_k) + sum(|constr|)     76.800001535686263000
                                              f(x_0)      6.802639150498489300

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

FuncEv    1 ConstrEv   30 ConJacEv   30 Iter   21 MinorIter  370
CPU time: 0.718750 sec. Elapsed time: 0.734000 sec. 

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Free Floating Robot 6c         f_k       4.160935929052688400
                                       sum(|constr|)      0.000000006428671067
                              f(x_k) + sum(|constr|)      4.160935935481359300
                                              f(x_0)      5.000000000000000000

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

FuncEv    1 ConstrEv   67 ConJacEv   67 Iter   27 MinorIter  749
CPU time: 0.859375 sec. Elapsed time: 0.859000 sec. 

end

Plot result

tf2 = tf_init;
tf3 = subs(tf,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 = 4.161