MediaWiki API result

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            "358": {
                "pageid": 358,
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                "title": "Robot arm problem",
                "revisions": [
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                        "*": "{{Dimensions\n\t\t|nd        = 1\n\t\t|nx        = 3\n\t\t|nu        = 3\n\t\t|nc        = 12\n\t\t|nre       = 12\n\t}}<!-- Do not insert line break here or Dimensions Box moves up in the layout...\n\t\n-->The robot arm problem focuses on minimizing the time used by a robot arm to move from an origin to a destination.\nThe arm is a bar of length <math> L </math> and sticks out distance <math> \\rho </math> from its moving axis, while sticking out distance <math> L - \\rho </math> in the other direction. The problem can be found in <bib id=\"Moessner1995\" /> or in the [http://www.mcs.anl.gov/~more/cops/ COPS library].\n\t\n== Model formulation ==\n\nThe problem is set up using the length <math> \\rho </math>, \"the vertical angles <math> (\\theta, \\Phi) </math> from the horizontal plane, the controls <math> u=(u_{\\rho},u_{\\theta},u_{\\Phi}) </math> and the final time <math> t_f </math>\".\n\t\nThe moving robot is modelled with the following equations:\n\t\n<math> \\ddot{\\rho} = \\frac{u_{\\rho}}{L}, \\qquad \\ddot{\\theta} = \\frac{u_{\\theta}}{I_{\\theta}}, \\qquad \\ddot{\\Phi} = \\frac{u_{\\Phi}}{I_{\\Phi}}</math>\n\t\t\t\nwhere <math> I </math> characterizes the moment of inertia, i.e.\n\t\n<math> \n\\begin{array}{ccl}\n\tI_{\\theta}  & = & \\frac{((L-\\rho)^3 + \\rho^3)}{3} \\cdot \\sin(\\Phi)^2, \\\\\n\tI_{\\Phi} & = & \\frac{((L-\\rho)^3 + \\rho^3)}{3}.\n\\end{array}\n</math>\n\nThe path constraints on the states <math> x= (\\rho, \\theta, \\Phi) </math> and on the controls <math> u = (u_{\\rho},u_{\\theta},u_{\\Phi}) </math> as well as the boundary conditions can be seen in the optimization problem further down.\n\t\n== Optimization problem ==\n\t\n<p>\n<math>\n\\begin{array}{llclr}\n\t\\displaystyle \\min_{x, u, t_f} & t_f   \\\\[1.5ex]\n\t\\mbox{s.t.} \n\t& \\ddot{\\rho} & = &  \\frac{u_{\\rho}}{L}, \\\\\n\t& \\ddot{\\theta} & = & \\frac{u_{\\theta}}{I_{\\theta}},  \\\\\n\t& \\ddot{\\Phi} & = & \\frac{u_{\\Phi}}{I_{\\Phi}},  \\\\[1.5ex]\n\t& x(0) &=& (4.5, 0, \\frac{\\pi}{4})^T, \\\\\n\t& x(t_f) &=& (4.5, \\frac{2\\pi}{3}, \\frac{\\pi}{4})^T, \\\\\n\t& \\dot{x}(0) &=& (0,0,0)^T, \\\\\n\t& \\dot{x}(t_f) &=& (0,0,0)^T, \\\\[1.5ex]\n\t& \\rho(t) & \\in & [0,L],\\\\\n\t& \\theta(t) & \\in & [-\\pi, \\pi],\\\\\n\t& \\Phi(t) & \\in & [0, \\pi],\\\\\n\t& u_{\\rho} & \\leq & 1,\\\\\n\t& u_{\\theta} & \\leq & 1,\\\\\n\t& u_{\\Phi} & \\leq & 1.\\\\\n\\end{array} \n</math>\n</p>\n\nwhere <math> I </math> is the moment of inertia as above.\n\t\n== Source Code ==\n\t\nModel descriptions are available in\n\n* [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Robot arm problem (TACO)]]\n\n\n\t== References ==\n\t<biblist />\n\t\n\t<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here -->\n\t[[Category:MIOCP]]\n\t[[Category:ODE model]]\n\t[[Category:Minimum time]]"
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            "167": {
                "pageid": 167,
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                "title": "Robot arm problem (TACO)",
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                        "*": "This page contains a model of the [[Robot arm problem]] in [http://www.ampl.org AMPL] format, making use of the TACO toolkit for AMPL control optimization extensions. This problem is due to <bib id=\"Moessner1995\" />. The original model using a collocation formulation can be found in the [http://www.mcs.anl.gov/~more/cops/ COPS library].\nNote that you will need to include a generic [[support AMPL files|AMPL/TACO support file]], OptimalControl.mod.\nTo solve this model, you require an optimal control or NLP code that uses the TACO toolkit to support the AMPL optimal control extensions.\n\n=== AMPL ===\n\nThis is the source file robotarm_taco.mod\n<source lang=\"AMPL\">\n# ----------------------------------------------------------------\n# Robot arm problem using AMPL and TACO\n# (c) Christian Kirches, Sven Leyffer\n#\n# Source: COPS 3.1 collocation formulation - March 2004\n#         David Bortz - Summer 1998          \n# ----------------------------------------------------------------\ninclude OptimalControl.mod;\n\n# Final positions of the length and the angles for the robot arm\n\nparam L;\t\t# total length of arm\nparam pi := 4*atan(1);\n\n# Upper bounds on the controls\n\nparam max_u_rho;\nparam max_u_the;\nparam max_u_phi;\n\n# Initial positions of the length and the angles for the robot arm\n\nparam rho_0;\nparam the_0;\nparam phi_0;\n\n# Final positions of the length and the angles for the robot arm\n\nparam rho_n;\nparam the_n;\nparam phi_n;\n\n# The length and the angles theta and phi for the robot arm.\n\nvar rho >=0, <= L; \nvar the >= -pi, <= pi; \nvar phi >=0, <= pi;\n\n# The derivatives of the length and the angles.\n    \nvar rho_dot;\nvar the_dot;\nvar phi_dot;\n\n# The controls.\n\nvar u_rho >= -max_u_rho, <= max_u_rho, suffix type \"u0\";\nvar u_the >= -max_u_the, <= max_u_the, suffix type \"u0\";\nvar u_phi >= -max_u_phi, <= max_u_phi, suffix type \"u0\";\n\n# The independent time and the final time.\n\nvar t;\nvar tf := 1.0, >= 0.1, <= 12 suffix scale 10.0;\n\n# The moments of inertia.\n\nvar I_the = ((L-rho)^3+rho^3)*(sin(phi))^2/3.0;\nvar I_phi = ((L-rho)^3+rho^3)/3.0;\n\n# The robot arm problem.\n\nminimize time: eval(t,tf) suffix scale 1e+2;\n\nsubject to rho_eqn:\n        diff(rho, t) = rho_dot;\n\nsubject to the_eqn:\n        diff(the, t) = the_dot;\n\nsubject to phi_eqn:\n        diff (phi, t) = phi_dot;\n\nsubject to u_rho_eqn:\n        diff (rho_dot, t) = u_rho/L;\n\nsubject to u_the_eqn:\n        diff (the_dot, t) = u_the/I_the;\n\nsubject to u_phi_eqn:\n        diff (phi_dot, t) = u_phi/I_phi;\n\n# Boundary Conditions\n\nsubject to rho_0_eqn: eval (rho, 0) = 4.5 suffix type \"dpc\";\nsubject to the_0_eqn: eval (the, 0) = 0 suffix type \"dpc\";\nsubject to phi_0_eqn: eval (phi, 0) = pi/4 suffix type \"dpc\";\n\nsubject to rho_f_eqn: eval (rho, tf) = 4.5;\nsubject to the_f_eqn: eval (the, tf) = 2*pi/3;\nsubject to phi_f_eqn: eval (phi, tf) = pi/4;\n\nsubject to rho_dot_0_eqn: eval (rho_dot, 0) = 0 suffix type \"dpc\";\nsubject to the_dot_0_eqn: eval (the_dot, 0) = 0 suffix type \"dpc\";\nsubject to phi_dot_0_eqn: eval (phi_dot, 0) = 0 suffix type \"dpc\";\n\nsubject to rho_dot_f_eqn: eval (rho_dot, tf) = 0;\nsubject to the_dot_f_eqn: eval (the_dot, tf) = 0;\nsubject to phi_dot_f_eqn: eval (phi_dot, tf) = 0;\n\noption solver ...;\n\nsolve;\n\n</source>\n\n== Other Descriptions ==\n\nOther descriptions of this problem are available in\n\n* Mathematical notation at [[Robot arm problem]]\n* [[:Category:AMPL | AMPL]] (using a fixed discretization) at the [http://www.mcs.anl.gov/~more/cops/ COPS library]\n\n== References ==\n<biblist />\n \n[[Category:AMPL/TACO]]"
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