# 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.

## 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'); 