PROPT Robot Arm Movement

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

Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY

Problem Formulation

Find u(t) over t in [0; 1 ] to minimize

$J=t_{f}$

subject to:

$L*{\frac {d^{2}rho}{dt^{2}}}=u_{1}$ $I_{1}*{\frac {d^{2}theta}{dt^{2}}}=u_{2}$ $I_{2}*{\frac {d^{2}phi}{dt^{2}}}=u_{3}$

$0<=rho<=L$ $|theta|<=pi$ $0<=phi<=pi$ $|u_{1:3}|<=1$

$I_{1}={\frac {((L-rho)^{3}+rho^{3})}{3}}*sin(phi)^{2}$ $I_{2}={\frac {((L-rho)^{3}+rho^{3})}{3}}$

The boundary conditions are:

$[rho_{0}\ theta_{0}\ phi_{0}]=[4.5\ 0\ {\frac {pi}{4}}]$ $[rho_{1}\ theta_{1}\ phi_{1}]=[4.5\ {\frac {2*pi}{3}}\ {\frac {pi}{4}}]$ $L=5$

All first order derivatives are 0 at boundaries.

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

Problem setup

toms t
toms t_f

% Initial guess
tfopt = 1;
x1opt = 4.5;
x2opt = 0;
x3opt = 2*pi/3*t.^2;
x4opt = 0;
x5opt = pi/4;
x6opt = 0;
u1opt = 0;
u2opt = 0;
u3opt = 0;

for n=[20 100]

    %rho d(rho)dt theta d(theta)dt phi d(phi)dt
    p = tomPhase('p', t, 0, t_f, n);
    setPhase(p);
    tomStates x1 x2 x3 x4 x5 x6
    tomControls u1 u2 u3

    % Initial guess
    x0 = {t_f == tfopt
        icollocate({x1 == x1opt
        x2 == x2opt; x3 == x3opt
        x4 == x4opt; x5 == x5opt
        x6 == x6opt})
        collocate({u1 == u1opt
        u2 == u2opt; u3 == u3opt})};

    % Box constraints
    L = 5;
    cbox = {
        0.1 <= t_f <= 10
        0   <= icollocate(x1) <= L
        -pi <= icollocate(x3) <= pi
        0   <= icollocate(x5) <= pi
        -1  <= collocate(u1)  <= 1
        -1  <= collocate(u2)  <= 1
        -1  <= collocate(u3)  <= 1};

    % Boundary constraints
    cbnd = {initial({x1 == 4.5; x2 == 0
        x3 == 0; x4 == 0
        x5 == pi/4; x6 == 0})
        final({x1 == 4.5; x2 == 0
        x3 == 2*pi/3
        x4 == 0
        x5 == pi/4
        x6 == 0
        })};

    I1 = ((L-x1).^3+x1.^3)./3.*sin(x5).^2;
    I2 = ((L-x1).^3+x1.^3)/3;

    % ODEs and path constraints
    ceq = collocate({dot(x1) == x2
        dot(x2) == u1/L;   dot(x3) == x4
        dot(x4) == u2./I1; dot(x5) == x6
        dot(x6) == u3./I2});

    % Objective
    objective = t_f;

Solve the problem

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

    % Optimal x, y, and speed, to use as starting guess in the next iteration
    x1opt = subs(x1, solution);
    x2opt = subs(x2, solution);
    x3opt = subs(x3, solution);
    x4opt = subs(x4, solution);
    x5opt = subs(x5, solution);
    u1opt = subs(u1, solution);
    u2opt = subs(u2, solution);
    u3opt = subs(u3, solution);
    tfopt = subs(final(t), solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.3121 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Robot Arm                      f_k       9.146367546080650600
                                       sum(|constr|)      0.000000278037569148
                              f(x_k) + sum(|constr|)      9.146367824118220500
                                              f(x_0)      1.000000000000000000

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

FuncEv    1 ConstrEv   17 ConJacEv   17 Iter   11 MinorIter  225
CPU time: 0.031200 sec. Elapsed time: 0.037000 sec. 

Problem type appears to be: lpcon
Time for symbolic processing: 0.3096 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Robot Arm                      f_k       9.140854009735493300
                                       sum(|constr|)      0.000002639179211847
                              f(x_k) + sum(|constr|)      9.140856648914704500
                                              f(x_0)      9.146367546080650600

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

FuncEv    1 ConstrEv    7 ConJacEv    7 Iter    4 MinorIter  729
CPU time: 1.606810 sec. Elapsed time: 0.634000 sec. 

end

t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x3 = subs(collocate(x3),solution);
x5 = subs(collocate(x5),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);
u3 = subs(collocate(u3),solution);

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x3,'*-',t,x5,'*-');
legend('rho','theta','phi');
title('Robot Arm state variables');

subplot(2,1,2)
plot(t,u1,'*-',t,u2,'*-',t,u3,'*-');
legend('u1','u2','u3');
title('Robot Arm control variables');