## 94  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.

### 94.1  Problem Formulation

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

 J = tF

subject to:

 x(0)  = [0  −2  0  0]
 x(tF) = [1  −1  0  0]

 L1 = 0.4;
 L2 = 0.4;
 m1 = 0.5;
 m2 = 0.5;
 Eye1 = 0.1;
 Eye2 = 0.1;
 el1 = 0.2;
 el2 = 0.2;

 cos(x2) = cos(x2);
 H11 = Eye1 + Eye2 + m1*el12 + m2*(L12+el22+2.0*L1*el2*cos(x2));
 H12 = Eye2 + m2*el22 + m2*L1*el2*cos(x2);
 H22 = Eye2 + m2*el22;
 h = m2*L1*el2*sin(x2);
delta =
 1.0 H11*H22−H12*H12

 dx1 dt
= x3
 dx2 dt
= x4
 dx3 dt
= delta*(2.0*h*H22*x3*x4+h*H22*x42 + h*H12*x32+H22*u1H12*u2);
 dx4 dt
= delta*(−2.0*h*H12*x3*x4h*H11*x32− h*H12*x42+H11*u2H12*u1);

 −10 <= u <= 10

Reference: [16]

### 94.2  Problem setup

```toms t
toms t_f

tfopt = 7;
x1opt = 1*t/t_f;
x2opt = -2+1*t/t_f;
x3opt = 2;
x4opt = 4;
u1opt = 10-20*t/t_f;
u2opt = -10+20*t/t_f;
```

### 94.3  Solve the problem, using a successively larger number collocation points

```for n=[30 60]
```
```    p = tomPhase('p', t, 0, t_f, n);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u1 u2

% Initial guess
x0 = {t_f == tfopt
icollocate({
x1 == x1opt
x2 == x2opt
x3 == x3opt
x4 == x4opt})
collocate({
u1 == u1opt
u2 == u2opt})};

% Box constraints
cbox = {
0.1 <= t_f <= 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;

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*cos(x2));
H_12  = Eye_2 + m_2*el_2^2 + m_2*L_1*el_2*cos(x2);
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 = t_f;
```

### 94.4  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
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Robot Manipulators             f_k       0.391698237386178260
sum(|constr|)      0.000063099634834817
f(x_k) + sum(|constr|)      0.391761337021013070
f(x_0)      7.000000000000000000

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

FuncEv    1 ConstrEv   68 ConJacEv   68 Iter   26 MinorIter  437
CPU time: 0.421875 sec. Elapsed time: 0.437000 sec.
```
```Problem type appears to be: lpcon
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2011-02-05
=====================================================================================
Problem: ---  1: Robot Manipulators             f_k       0.391820155673056890
sum(|constr|)      0.000000000017635038
f(x_k) + sum(|constr|)      0.391820155690691950
f(x_0)      0.391698237386178260

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

FuncEv    1 ConstrEv   16 ConJacEv   16 Iter   12 MinorIter  440
CPU time: 0.515625 sec. Elapsed time: 0.547000 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);
```

### 94.5  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');
```