Reference
Category : GAMS NOA library
Mainfile : robot.gms
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Minimize the time taken for a robot arm to travel between two points.
This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.
References:
* Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS.
Tech. rep., Mathematics and Computer Science Division, 2000.
* Vanderbei, R, Nonlinear Optimization Models.
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$if set n $set nh %n%
$if not set nh $set nh 800
sets h intervals / h0 * h%nh%/
scalars
pi
nh number of intervals / %nh% /
L total length of arm / 5 /
max_u_rho / 1 /
max_u_the / 1 /
max_u_phi / 1 /;
pi = 2*arctan(inf);
variables
rho(h) distance from the origin
the(h) horizontal angle
phi(h) vertical angle
rho_dot(h)
the_dot(h)
phi_dot(h)
u_rho(h) control
u_the(h)
u_phi(h)
step
tf final time
i_the(h) moment of inertia
i_phi(h) ;
* Bounds
rho.lo(h) = 0; rho.up(h) = L;
the.lo(h) = -pi; the.up(h) = pi;
phi.lo(h) = 0; phi.up(h) = pi;
u_rho.lo(h) = -max_u_rho; u_rho.up(h) = max_u_rho;
u_the.lo(h) = -max_u_the; u_the.up(h) = max_u_the;
u_phi.lo(h) = -max_u_phi; u_phi.up(h) = max_u_phi;
i_the.lo(h) = 0.0001;
i_phi.lo(h) = 0.0001;
set firstlast(h) / h0, h%nh% /;
* Fixed variables
the.fx('h0') = 0;
the.fx('h%nh%') = 2*pi/3;
rho.fx(firstlast) = 4.5;
phi.fx(firstlast) = pi/4;
rho_dot.fx(firstlast) = 0;
the_dot.fx(firstlast) = 0;
phi_dot.fx(firstlast) = 0;
i_phi.fx(firstlast(h)) = (power(L-rho.l(h),3)+power(rho.l(h),3))/3.0;
i_the.fx(firstlast(h)) = i_phi.l(h)*sqr(sin(phi.l(h)));
*Initialization
rho.l(h) = 4.5;
the.l(h) = (2*pi/3)*sqr(ord(h)/nh);
phi.l(h) = pi/4;
rho_dot.l(h) = 0.0;
the_dot.l(h) = (4*pi/3)*(ord(h)/nh);
phi_dot.l(h) = 0.0;
step.l = 1/nh;
i_phi.l(h) = (power(L-rho.l(h),3)+power(rho.l(h),3))/3.0;
i_the.l(h) = i_phi.l(h)*sqr(sin(phi.l(h)));
equations
tf_eqn
rho_eqn(h)
the_eqn(h)
phi_eqn(h)
u_rho_eqn(h)
u_the_eqn(h)
u_phi_eqn(h)
i_the_eqn(h)
i_phi_eqn(h);
tf_eqn.. tf =e= step*nh;
i_phi_eqn(h).. i_phi(h) =e= (power(L-rho(h),3)+power(rho(h),3))/3.0;
i_the_eqn(h).. i_the(h) =e= i_phi(h)*sqr(sin(phi(h)));
rho_eqn(h-1).. rho(h) =e= rho(h-1) + 0.5*step*(rho_dot(h) + rho_dot(h-1));
the_eqn(h-1).. the(h) =e= the(h-1) + 0.5*step*(the_dot(h) + the_dot(h-1));
phi_eqn(h-1).. phi(h) =e= phi(h-1) + 0.5*step*(phi_dot(h) + phi_dot(h-1));
u_rho_eqn(h-1).. rho_dot(h) =e=
rho_dot(h-1) + 0.5*step*(u_rho(h) + u_rho(h-1))/L;
u_the_eqn(h-1).. the_dot(h) =e=
the_dot(h-1) + 0.5*step*(u_the(h)/i_the(h) + u_the(h-1)/i_the(h-1));
u_phi_eqn(h-1).. phi_dot(h) =e=
phi_dot(h-1) + 0.5*step*(u_phi(h)/i_phi(h) + u_phi(h-1)/i_phi(h-1));
model robot /all/;
$ifThenI x%mode%==xbook
robot.iterlim=50000;
robot.workspace=120;
$endIf
solve robot miniziming tf using nlp;
$ifThenI x%mode%==xbook
file res /robot.dat/;
put res
loop(h, put u_phi.l(h):10:5, put/)
$endIf
*-------------------------- Numerical Experiments--------------------
* January 15, 2011
*
*Variant 1:
* 8002 constraints, 11012 variables
* CONOPT3:
* 953 iterations, 27.720 seconds
* vfo=9.1409312779
* KNITRO:
* Time limit reached
*--------------------------------------------------------------------
* End Robot