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robot.gms : Robot arm COPS 2.0 #8

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/.

The number of discretization points can be specified using the command
line parameter --nh. COPS performance tests have been reported for nh
= 50, 100, 200, 400
References:
Large Model of Type: NLP
$Title Robot arm COPS 2.0 #8 (ROBOT,SEQ=236) $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/. The number of discretization points can be specified using the command line parameter --nh. COPS performance tests have been reported for nh = 50, 100, 200, 400 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 50 sets h intervals / h0 * h%nh%/ scalars pi a famous constant 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) the(h) phi(h) rho_dot(h) the_dot(h) phi_dot(h) u_rho(h) u_the(h) u_phi(h) step tf i_the(h) i_phi(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)); 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% /; 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))); 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))); model robot /all/; $if set workspace robot.workspace = %workspace%; solve robot miniziming tf using nlp;