RobotArm

This problem can be found here. although that example is missing initial and final state constraints and limits on x4

Packages that will be used

using NLOptControl

Differential Equations

const EP=2*eps()
function RobotArm{T<:Any}(n::NLOpt,x::Array{T,2},u::Array{T,2}) # dynamic constraint equations
  if n.s.integrationMethod==:tm; L=size(x)[1]; else; L=size(x)[1]-1; end
  dx=Array(Any,L,n.numStates);
  Q=5;
  I_t=@NLexpression(n.mdl, [j=1:L], ((Q-x[j,1])^3+x[j,1]^3)/3*sin(x[j,5])^2 );
  I_p=@NLexpression(n.mdl, [j=1:L], ((Q-x[j,1])^3+x[j,1]^3)/3 );

  dx[:,1]=@NLexpression(n.mdl, [j=1:L], x[j,2]);
  dx[:,2]=@NLexpression(n.mdl, [j=1:L], u[j,1]/Q);
  dx[:,3]=@NLexpression(n.mdl, [j=1:L], x[j,4]);
  dx[:,4]=@NLexpression(n.mdl, [j=1:L], u[j,2]/(I_t[j]+EP));
  dx[:,5]=@NLexpression(n.mdl, [j=1:L], x[j,6]);
  dx[:,6]=@NLexpression(n.mdl, [j=1:L], u[j,3]/(I_p[j]+EP));
  return dx
end

Define and Configure the Problem:

n=define!(RobotArm;numStates=6,numControls=3,X0=[9/2,0.0,0.0,0.0,pi/4,0.0],XF=[9/2,0.0,2*pi/3,0.0,pi/4,0.0],XL=[NaN,NaN,NaN,0.0,NaN,NaN],XU=[NaN,NaN,NaN,1.0,NaN,NaN],CL=[-1.,-1.,-1.],CU=[1.,1.,1.])
configure!(n;(:finalTimeDV=>true))

Objective Function

@NLobjective(n.mdl,Min,n.tf);

Optimize

optimize!(n);


******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
 Ipopt is released as open source code under the Eclipse Public License (EPL).
         For more information visit http://projects.coin-or.org/Ipopt
******************************************************************************

Post Process

using PrettyPlots
allPlots(n)