PROPT Path Tracking Robot (Two-Phase) - Revision history

Per: /* Plot result */

2012-02-14T09:32:26Z

Plot result

← Older revision		Revision as of 04:32, 14 February 2012
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	[[File:pathTrackingRobot_twophase_01.png]]		[[File:pathTrackingRobot_twophase_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:11, 9 November 2011

2011-11-09T12:11:15Z

Mbot at 13:07, 8 November 2011

2011-11-08T13:07:31Z

← Older revision		Revision as of 08:07, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: qp		Problem type appears to be: qp
	Time for symbolic processing: 0.~~27638~~ seconds		Time for symbolic processing: 0.25294 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	Iter 297		Iter 297
	CPU time: 0.046800 sec. Elapsed time: 0.~~092000~~ sec.		CPU time: 0.046800 sec. Elapsed time: 0.123000 sec.

	</pre>		</pre>

Mbot at 18:25, 2 November 2011

2011-11-02T18:25:40Z

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	{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}		{{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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	title('Path Tracking Robot control variables');		title('Path Tracking Robot control variables');
	</source>		</source>

			[[File:pathTrackingRobot_twophase_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} User's Guide for DIRCOL 2.7 Optimal path tracking for a simple robot. A robot with two rotational joint..."

2011-11-02T15:30:28Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} User's Guide for DIRCOL 2.7 Optimal path tracking for a simple robot. A robot with two rotational joint..."

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{{Part Of Manual|title=the PROPT Manual|link=[[PROPT |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

<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:

<math> \frac{d^2q_1}{dt^2} = u_1 </math>
<math> \frac{d^2q_2}{dt^2} = u_2 </math>

A transformation gives:

<math> \frac{dx_1}{dt} = x_3 </math>
<math> \frac{dx_2}{dt} = x_4 </math>
<math> \frac{dx_3}{dt} = u_1 </math>
<math> \frac{dx_4}{dt} = u_2 </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> 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_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_4{1,ref} = 0 \ (0<t<1), \ \frac{1}{2} \ (1<t<2) </math>

<math> |u| < 10 </math>

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
t_F = 2;
toms t1
p1 = tomPhase('p1', t1, 0, t_F/2, 25);
toms t2
p2 = tomPhase('p2', t2, t_F/2, t_F/2, 25);
setPhase(p1);

tomStates x1p1 x2p1 x3p1 x4p1
tomControls u1p1 u2p1

% Box constraints
cbox1 = {-10 <= collocate(u1p1) <= 10
-10 <= collocate(u2p1) <= 10};

% Boundary constraints
w1 = 100; w2 = 100;
w3 = 500; w4 = 500;
err1p1 = w1*(x1p1-t1/2).^2;
err2p1 = w2*(x2p1).^2;
err3p1 = w3*(x3p1-1/2).^2;
err4p1 = w4*(x4p1).^2;

cbnd1 = initial({x1p1 == 0; x2p1 == 0
x3p1 == 0.5; x4p1 == 0});

% ODEs and path constraints
ceq1 = collocate({dot(x1p1) == x3p1
dot(x2p1) == x4p1; dot(x3p1) == u1p1
dot(x4p1) == u2p1});

% Objective
objective1 = integrate(err1p1+err2p1+err3p1+err4p1);

% Phase 2
setPhase(p2);
tomStates x1p2 x2p2 x3p2 x4p2
tomControls u1p2 u2p2

% Box constraints
cbox2 = {-10 <= collocate(u1p2) <= 10
-10 <= collocate(u2p2) <= 10};

% Boundary constraints
err1p2 = w1*(x1p2-1/2).^2;
err2p2 = w2*(x2p2-(t2-1)/2).^2;
err3p2 = w3*(x3p2).^2;
err4p2 = w4*(x4p2-1/2).^2;

cbnd2 = final({x1p2 == 0.5
x2p2 == 0.5
x3p2 == 0
x4p2 == 0.5});

% ODEs and path constraints
ceq2 = collocate({
dot(x1p2) == x3p2
dot(x2p2) == x4p2
dot(x3p2) == u1p2
dot(x4p2) == u2p2});

% Objective
objective2 = integrate(err1p2+err2p2+err3p2+err4p2);

% Objective
objective = objective1 + objective2;

% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)
final(p1,x2p1) == initial(p2,x2p2)
final(p1,x3p1) == initial(p2,x3p2)
final(p1,x4p1) == initial(p2,x4p2)};
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Path Tracking Robot (Two-Phase)';
options.solver = 'sqopt7';
constr = {cbox1, cbnd1, ceq1, cbox2, cbnd2, ceq2, link};
solution = ezsolve(objective, constr, [], options);
t = subs(collocate(p1,t1),solution);
t = [t;subs(collocate(p2,t2),solution)];
x1 = subs(collocate(p1,x1p1),solution);
x1 = [x1;subs(collocate(p2,x1p2),solution)];
x2 = subs(collocate(p1,x2p1),solution);
x2 = [x2;subs(collocate(p2,x2p2),solution)];
x3 = subs(collocate(p1,x3p1),solution);
x3 = [x3;subs(collocate(p2,x3p2),solution)];
x4 = subs(collocate(p1,x4p1),solution);
x4 = [x4;subs(collocate(p2,x4p2),solution)];
u1 = subs(collocate(p1,u1p1),solution);
u1 = [u1;subs(collocate(p2,u1p2),solution)];
u2 = subs(collocate(p1,u2p1),solution);
u2 = [u2;subs(collocate(p2,u2p2),solution)];
</source>

<pre>
Problem type appears to be: qp
Time for symbolic processing: 0.27638 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: 1: Path Tracking Robot (Two-Phase) f_k 1.049952991376699200
sum(|constr|) 0.000000000261665098
f(x_k) + sum(|constr|) 1.049952991638364300
f(x_0) 0.000000000000000000

Solver: SQOPT. EXIT=0. INFORM=1.
SQOPT 7.2-5 QP solver
Optimality conditions satisfied

Iter 297
CPU time: 0.046800 sec. Elapsed time: 0.092000 sec.

</pre>

==Plot result==

<source lang="matlab">
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');
</source>

← Older revision		Revision as of 07:11, 9 November 2011
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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">