https://mintoc.de/index.php?action=history&feed=atom&title=Robot_arm_problemRobot arm problem - Revision history2026-06-09T09:09:15ZRevision history for this page on the wikiMediaWiki 1.43.1https://mintoc.de/index.php?title=Robot_arm_problem&diff=2140&oldid=prevFelixMueller: Created page with "{{Dimensions |nd = 1 |nx = 3 |nu = 3 |nc = 12 |nre = 12 }}<!-- Do not insert line break here or Dimensions Box moves up in the lay..."2016-07-31T07:15:48Z<p>Created page with "{{Dimensions |nd = 1 |nx = 3 |nu = 3 |nc = 12 |nre = 12 }}<!-- Do not insert line break here or Dimensions Box moves up in the lay..."</p>
<p><b>New page</b></p><div>{{Dimensions<br />
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-->The robot arm problem focuses on minimizing the time used by a robot arm to move from an origin to a destination.<br />
The 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].<br />
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== Model formulation ==<br />
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The 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>".<br />
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The moving robot is modelled with the following equations:<br />
<br />
<math> \ddot{\rho} = \frac{u_{\rho}}{L}, \qquad \ddot{\theta} = \frac{u_{\theta}}{I_{\theta}}, \qquad \ddot{\Phi} = \frac{u_{\Phi}}{I_{\Phi}}</math><br />
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where <math> I </math> characterizes the moment of inertia, i.e.<br />
<br />
<math> <br />
\begin{array}{ccl}<br />
I_{\theta} & = & \frac{((L-\rho)^3 + \rho^3)}{3} \cdot \sin(\Phi)^2, \\<br />
I_{\Phi} & = & \frac{((L-\rho)^3 + \rho^3)}{3}.<br />
\end{array}<br />
</math><br />
<br />
The 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.<br />
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== Optimization problem ==<br />
<br />
<p><br />
<math><br />
\begin{array}{llclr}<br />
\displaystyle \min_{x, u, t_f} & t_f \\[1.5ex]<br />
\mbox{s.t.} <br />
& \ddot{\rho} & = & \frac{u_{\rho}}{L}, \\<br />
& \ddot{\theta} & = & \frac{u_{\theta}}{I_{\theta}}, \\<br />
& \ddot{\Phi} & = & \frac{u_{\Phi}}{I_{\Phi}}, \\[1.5ex]<br />
& x(0) &=& (4.5, 0, \frac{\pi}{4})^T, \\<br />
& x(t_f) &=& (4.5, \frac{2\pi}{3}, \frac{\pi}{4})^T, \\<br />
& \dot{x}(0) &=& (0,0,0)^T, \\<br />
& \dot{x}(t_f) &=& (0,0,0)^T, \\[1.5ex]<br />
& \rho(t) & \in & [0,L],\\<br />
& \theta(t) & \in & [-\pi, \pi],\\<br />
& \Phi(t) & \in & [0, \pi],\\<br />
& u_{\rho} & \leq & 1,\\<br />
& u_{\theta} & \leq & 1,\\<br />
& u_{\Phi} & \leq & 1.\\<br />
\end{array} <br />
</math><br />
</p><br />
<br />
where <math> I </math> is the moment of inertia as above.<br />
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== Source Code ==<br />
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Model descriptions are available in<br />
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* [[:Category:AMPL/TACO | AMPL/TACO code]] at [[Robot arm problem (TACO)]]<br />
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== References ==<br />
<biblist /><br />
<br />
<!--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 --><br />
[[Category:MIOCP]]<br />
[[Category:ODE model]]<br />
[[Category:Minimum time]]</div>FelixMueller