PROPT Path Tracking Robot: Difference between revisions
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{{Part Of Manual|title=the PROPT | {{Part Of Manual|title=the PROPT Manual|link=[[PROPT|PROPT Manual]]}} | ||
User's Guide for DIRCOL | User's Guide for DIRCOL | ||
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<math> 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 </math> | <math> 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 </math> | ||
subject to: | subject to: | ||
<math> \frac{d^2q_1}{dt^2} = u_1 </math> | <math> \frac{d^2q_1}{dt^2} = u_1 </math> | ||
<math> \frac{d^2q_2}{dt^2} = u_2 </math> | <math> \frac{d^2q_2}{dt^2} = u_2 </math> | ||
A transformation gives: | A transformation gives: | ||
<math> \frac{dx_1}{dt} = x_3 </math> | <math> \frac{dx_1}{dt} = x_3 </math> | ||
<math> \frac{dx_2}{dt} = x_4 </math> | <math> \frac{dx_2}{dt} = x_4 </math> | ||
<math> \frac{dx_3}{dt} = u_1 </math> | <math> \frac{dx_3}{dt} = u_1 </math> | ||
<math> \frac{dx_4}{dt} = u_2 </math> | <math> \frac{dx_4}{dt} = u_2 </math> | ||
<math> x_{1:4}(0) = [0 \ 0 \ 0.5 \ 0] </math> | <math> x_{1:4}(0) = [0 \ 0 \ 0.5 \ 0] </math> | ||
<math> x_{1:4}(2) = [0.5 \ 0.5 \ 0 \ 0.5] </math> | <math> x_{1:4}(2) = [0.5 \ 0.5 \ 0 \ 0.5] </math> | ||
<math> w_{1:4} = [100 \ 100 \ 500 \ 500] </math> | <math> w_{1:4} = [100 \ 100 \ 500 \ 500] </math> | ||
<math> x_1{1,ref} = \frac{t}{2} \ (0<t<1), \ \frac{1}{2} \ (1<t<2) </math> | <math> x_1{1,ref} = \frac{t}{2} \ (0<t<1), \ \frac{1}{2} \ (1<t<2) </math> | ||
<math> x_2{1,ref} = 0 \ (0<t<1), \ \frac{t-1}{2} \ (1<t<2) </math> | <math> x_2{1,ref} = 0 \ (0<t<1), \ \frac{t-1}{2} \ (1<t<2) </math> | ||
<math> x_3{1,ref} = \frac{1}{2} \ (0<t<1), \ 0 \ (1<t<2) </math> | <math> x_3{1,ref} = \frac{1}{2} \ (0<t<1), \ 0 \ (1<t<2) </math> | ||
<math> x_4{1,ref} = 0 \ (0<t<1), \ \frac{1}{2} \ (1<t<2) </math> | <math> x_4{1,ref} = 0 \ (0<t<1), \ \frac{1}{2} \ (1<t<2) </math> | ||
<math> |u| < 10 </math> | <math> |u| < 10 </math> | ||
<source lang="matlab"> | <source lang="matlab"> | ||
Line 99: | Line 114: | ||
<pre> | <pre> | ||
Problem type appears to be: qp | Problem type appears to be: qp | ||
Time for symbolic processing: 0. | Time for symbolic processing: 0.16121 seconds | ||
Starting numeric solver | Starting numeric solver | ||
===== * * * =================================================================== * * * | ===== * * * =================================================================== * * * | ||
Line 114: | Line 129: | ||
FuncEv 10 GradEv 10 ConstrEv 10 Iter 10 | FuncEv 10 GradEv 10 ConstrEv 10 Iter 10 | ||
CPU time: 0. | CPU time: 0.249602 sec. Elapsed time: 0.061000 sec. | ||
</pre> | </pre> | ||
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title('Path Tracking Robot control variables'); | title('Path Tracking Robot control variables'); | ||
</source> | </source> | ||
[[File:pathTrackingRobot_01.png]] | |||
[[Category:PROPT Examples]] |
Latest revision as of 05:32, 14 February 2012
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
subject to:
A transformation gives:
% 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');