Robot : Minimize the Time Taken for a Robot Arm to Travel Between Two Points

Reference

  • Neculai Andrei, Nonlinear Optimization Applications Using the GAMS Technology,Springer Optimization and Its Applications, Model Robot (5.35) in chapter Applications of Mechanical Engineering , 2013

Category : GAMS NOA library


Mainfile : robot.gms

$ontext
     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.
$offtext


$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
$onecho >bench.opt
  solvers minos
$offecho

robot.optfile=1;
robot.iterlim=50000;
robot.workspace=120;

option nlp=bench
$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