PROPT Path Tracking Robot

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

User's Guide for DIRCOL

2.7 Optimal path tracking for a simple robot. A robot with two rotational joints and simplified equations of motion has to move along a prescribed path with constant velocity.

Problem Formulation

Find u over t in [0; 2 ] to minimize

$J=\int _{0}^{2}(\sum _{i=1}^{2}(w_{i}*(q_{i}(t)-q_{i,ref})^{2})+\sum _{i=1}^{2}(w_{2+i}*({\frac {dq}{dt}}_{i}(t)-{\frac {dq}{dt}}_{i,ref})^{2})\mathrm {d} t$

subject to:

${\frac {d^{2}q_{1}}{dt^{2}}}=u_{1}$

${\frac {d^{2}q_{2}}{dt^{2}}}=u_{2}$

A transformation gives:

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

${\frac {dx_{2}}{dt}}=x_{4}$

${\frac {dx_{3}}{dt}}=u_{1}$

${\frac {dx_{4}}{dt}}=u_{2}$

$x_{1:4}(0)=[0\ 0\ 0.5\ 0]$

$x_{1:4}(2)=[0.5\ 0.5\ 0\ 0.5]$

$w_{1:4}=[100\ 100\ 500\ 500]$

$x_{1}{1,ref}={\frac {t}{2}}\ (0<t<1),\ {\frac {1}{2}}\ (1<t<2)$

$x_{2}{1,ref}=0\ (0<t<1),\ {\frac {t-1}{2}}\ (1<t<2)$

$x_{3}{1,ref}={\frac {1}{2}}\ (0<t<1),\ 0\ (1<t<2)$

$x_{4}{1,ref}=0\ (0<t<1),\ {\frac {1}{2}}\ (1<t<2)$

$|u|<10$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 2, 100, [], 'fem1s'); % Use splines with FEM constraints
%p = tomPhase('p', t, 0, 2, 100, [], 'fem1');  % Use linear finite elements
%p = tomPhase('p', t, 0, 2, 100); % Use Gauss point collocation
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u1 u2

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

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

% ODEs and path constraints
w1 = 100; w2 = 100;
w3 = 500; w4 = 500;

err1 = w1*(x1-t/2.*(t<1)-1/2*(t>=1)).^2;
err2 = w2*(x2-(t-1)/2.*(t>=1)).^2;
err3 = w3*(x3-1/2*(t<1)).^2;
err4 = w4*(x4-1/2*(t>=1)).^2;

toterr = integrate(err1+err2+err3+err4);

ceq = collocate({
    dot(x1) == x3
    dot(x2) == x4
    dot(x3) == u1
    dot(x4) == u2});

% Objective
objective = toterr;

Solve the problem

options = struct;
options.name = 'Path Tracking Robot';
solution = ezsolve(objective, {cbox, cbnd, ceq}, [], options);
t  = subs(icollocate(t),solution);
x1 = subs(icollocate(x1),solution);
x2 = subs(icollocate(x2),solution);
x3 = subs(icollocate(x3),solution);
x4 = subs(icollocate(x4),solution);
u1 = subs(icollocate(u1),solution);
u2 = subs(icollocate(u2),solution);

Problem type appears to be: qp
Time for symbolic processing: 0.16121 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem:  1: Path Tracking Robot                f_k       1.031157512955303400
                                       sum(|constr|)      0.000000055682846425
                              f(x_k) + sum(|constr|)      1.031157568638149800
                                              f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Barrier QP solver
Optimal solution found

FuncEv   10 GradEv   10 ConstrEv   10 Iter   10 
CPU time: 0.249602 sec. Elapsed time: 0.061000 sec. 

Plot result

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

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