Time-optimal Trajectories for Robot Manipulators

Users Guide for dyn.Opt, Example 2

Dissanayake, M., Goh, C. J., & Phan-Thien, N., Time-optimal Trajectories for Robot Manipulators, Robotica, Vol. 9, pp. 131-138, 1991.

Contents

Problem Formulation

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

subject to:

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

Problem setup

toms t
toms tf

tfopt = 7;
x1opt = 1*t/tf;
x2opt = -2+1*t/tf;
x3opt = 2;
x4opt = 4;
u1opt = 10-20*t/tf;
u2opt = -10+20*t/tf;

Solve the problem, using a successively larger number collocation points

for n=[30 60]

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

    tomStates x1 x2 x3 x4
    tomControls u1 u2

    % Initial guess
    x0 = {tf == tfopt
        icollocate({
        x1 == x1opt
        x2 == x2opt
        x3 == x3opt
        x4 == x4opt})
        collocate({
        u1 == u1opt
        u2 == u2opt})};

    % Box constraints
    cbox = {
        0.1 <= tf <= 50
        -10 <= collocate(u1) <= 10
        -10 <= collocate(u2) <= 10};

    % Boundary constraints
    cbnd = {initial({x1 == 0; x2 == -2
        x3 == 0; x4 == 0})
        final({x1 == 1; x2 == -1
        x3 == 0; x4 == 0})};

    % ODEs and path constraints
    L_1 = 0.4;   L_2 = 0.4;
    m_1 = 0.5;   m_2 = 0.5;
    Eye_1 = 0.1; Eye_2 = 0.1;
    el_1 = 0.2;  el_2 = 0.2;

    cs1   = cos(x2);
    H_11  = Eye_1 + Eye_2 + m_1*el_1^2+ ...
        m_2*(L_1^2+el_2^2+2.0*L_1*el_2*cs1);
    H_12  = Eye_2 + m_2*el_2^2 + m_2*L_1*el_2*cs1;
    H_22  = Eye_2 + m_2*el_2^2;
    h     = m_2*L_1*el_2*sin(x2);
    delta = 1.0./(H_11.*H_22-H_12.^2);

    ceq = collocate({
        dot(x1) == x3
        dot(x2) == x4
        dot(x3) == delta.*(2.0*h.*H_22.*x3.*x4 ...
        +h.*H_22.*x4.^2+h.*H_12.*x3.^2+H_22.*u1-H_12.*u2)
        dot(x4) == delta.*(-2.0*h.*H_12.*x3.*x4 ...
        -h.*H_11.*x3.^2-h.*H_12.*x4.^2+H_11.*u2-H_12.*u1)});

    % Objective
    objective = tf;

Solve the problem

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

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

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Robot Manipulators             f_k       0.391698225312772320
                                       sum(|constr|)      0.000156851954993457
                              f(x_k) + sum(|constr|)      0.391855077267765750
                                              f(x_0)      7.000000000000000000

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

FuncEv    1 ConstrEv   51 ConJacEv   51 Iter   25 MinorIter  280
CPU time: 0.312500 sec. Elapsed time: 0.328000 sec. 

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Robot Manipulators             f_k       0.391820155673056890
                                       sum(|constr|)      0.000000000019002159
                              f(x_k) + sum(|constr|)      0.391820155692059020
                                              f(x_0)      0.391698225312772320

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

FuncEv    1 ConstrEv   16 ConJacEv   16 Iter   12 MinorIter  438
CPU time: 0.500000 sec. Elapsed time: 0.515000 sec. 

end

t  = subs(collocate(t),solution);
x1 = subs(collocate(x1opt),solution);
x2 = subs(collocate(x2opt),solution);
x3 = subs(collocate(x3opt),solution);
x4 = subs(collocate(x4opt),solution);
u1 = subs(collocate(u1opt),solution);
u2 = subs(collocate(u2opt),solution);

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Robot Manipulators state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Robot Manipulators control');