PROPT Two-Link Robotic Arm

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

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

12.4.2 Example 2: Two-link robotic arm

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Problem Formulation

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

$J=t_{F}$

subject to:

${\frac {dx_{1}}{dt}}={\frac {sin(x_{3})*({\frac {9}{4}}*cos(x_{3})*x_{1}^{2}+2*x_{2}^{2})+{\frac {4}{3}}*(u_{1}-u_{2})-{\frac {3}{2}}*cos(x_{3})*u_{2}}{{\frac {31}{36}}+{\frac {9}{4}}*sin(x_{3})^{2}}}$

${\frac {dx_{2}}{dt}}=-{\frac {sin(x_{3})*({\frac {7}{2}}*x_{1}^{2}+{\frac {9}{4}}*cos(x3)*x_{2}^{2})-{\frac {7}{3}}*u_{2}+{\frac {3}{2}}*cos(x_{3})*(u_{1}-u_{2})}{{\frac {31}{36}}+{\frac {9}{4}}*sin(x_{3})^{2}}}$

${\frac {dx_{3}}{dt}}=x_{2}-x_{1}$

${\frac {dx_{4}}{dt}}=x_{1}$

The initial condition are:

$x(0)=[0\ 0\ 0.5\ 0]$

$x(t_{F})=[0\ 0\ 0.5\ 0.522]$

$-1<=u(1:2)<=1$

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

Problem setup

toms t t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u1 u2

% Initial guess
x0 = {t_f == 3
    icollocate({x1 == 0; x2 == 0
    x3 == 0.5; x4 == 0.522})
    collocate({u1 == 1-2*t/t_f
    u2 == 1-2*t/t_f})};

% Box constraints
cbox = {2.6 <= t_f <= 100
    -1  <= collocate(u1) <= 1
    -1  <= collocate(u2) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0
    x3 == 0.5; x4 == 0})
    final({x1 == 0; x2 == 0
    x3 == 0.5; x4 == 0.522})};

% ODEs and path constraints
ceq = collocate({
    dot(x1) == ( sin(x3).*(9/4*cos(x3).*x1.^2+2*x2.^2) ...
    +4/3*(u1-u2) - 3/2*cos(x3).*u2 )./ (31/36 + 9/4*sin(x3).^2)
    dot(x2) == -( sin(x3).*(7/2*x1.^2 + 9/4*cos(x3).*x2.^2) ...
    - 7/3*u2 + 3/2*cos(x3).*(u1-u2) )./ (31/36 + 9/4*sin(x3).^2)
    dot(x3) == x2-x1
    dot(x4) == x1});

% Objective
objective = t_f;

Solve the problem

options = struct;
options.name = 'Two Link Robotic Arm';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);

Problem type appears to be: lpcon
Time for symbolic processing: 0.41192 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Two Link Robotic Arm           f_k       2.983364855223867600
                                       sum(|constr|)      0.000000154455553673
                              f(x_k) + sum(|constr|)      2.983365009679421300
                                              f(x_0)      3.000000000000000000

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

FuncEv    1 ConstrEv   20 ConJacEv   20 Iter   16 MinorIter  278
CPU time: 0.062400 sec. Elapsed time: 0.062000 sec. 

Plot result

subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Two Link Robotic Arm state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Two Link Robotic Arm control');