Robot arm problem

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Robot arm problem
State dimension: 1
Differential states: 3
Continuous control functions: 3
Path constraints: 12
Interior point equalities: 12

The robot arm problem focuses on minimizing the time used by a robot arm to move from an origin to a destination. The arm is a bar of length  L and sticks out distance  \rho from its moving axis, while sticking out distance  L - \rho in the other direction. The problem can be found in [Moessner1995]Address: Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
Author: M. Moessner-Beigel
Month: November
School: Ruprecht-Karls-Universit\"at Heidelberg
Title: Optimale Steuerung f\"ur Industrieroboter unter Ber\"ucksichtigung der getriebebedingten Elastizit\"at
Type: Diplomarbeit
Year: 1995
Link to Google Scholar
or in the COPS library.

Model formulation

The problem is set up using the length  \rho , "the vertical angles  (\theta, \Phi) from the horizontal plane, the controls  u=(u_{\rho},u_{\theta},u_{\Phi}) and the final time  t_f ".

The moving robot is modelled with the following equations:

 \ddot{\rho} = \frac{u_{\rho}}{L}, \qquad \ddot{\theta} = \frac{u_{\theta}}{I_{\theta}}, \qquad \ddot{\Phi} = \frac{u_{\Phi}}{I_{\Phi}}

where  I characterizes the moment of inertia, i.e.

 
\begin{array}{ccl}
	I_{\theta}  & = & \frac{((L-\rho)^3 + \rho^3)}{3} \cdot \sin(\Phi)^2, \\
	I_{\Phi} & = & \frac{((L-\rho)^3 + \rho^3)}{3}.
\end{array}

The path constraints on the states  x= (\rho, \theta, \Phi) and on the controls  u = (u_{\rho},u_{\theta},u_{\Phi}) as well as the boundary conditions can be seen in the optimization problem further down.

Optimization problem


\begin{array}{llclr}
	\displaystyle \min_{x, u, t_f} & t_f   \\[1.5ex]
	\mbox{s.t.} 
	& \ddot{\rho} & = &  \frac{u_{\rho}}{L}, \\
	& \ddot{\theta} & = & \frac{u_{\theta}}{I_{\theta}},  \\
	& \ddot{\Phi} & = & \frac{u_{\Phi}}{I_{\Phi}},  \\[1.5ex]
	& x(0) &=& (4.5, 0, \frac{\pi}{4})^T, \\
	& x(t_f) &=& (4.5, \frac{2\pi}{3}, \frac{\pi}{4})^T, \\
	& \dot{x}(0) &=& (0,0,0)^T, \\
	& \dot{x}(t_f) &=& (0,0,0)^T, \\[1.5ex]
	& \rho(t) & \in & [0,L],\\
	& \theta(t) & \in & [-\pi, \pi],\\
	& \Phi(t) & \in & [0, \pi],\\
	& u_{\rho} & \leq & 1,\\
	& u_{\theta} & \leq & 1,\\
	& u_{\Phi} & \leq & 1.\\
\end{array}

where  I is the moment of inertia as above.

Source Code

Model descriptions are available in


== References ==

[Moessner1995]M. Moessner-Beigel: Optimale Steuerung f\"ur Industrieroboter unter Ber\"ucksichtigung der getriebebedingten Elastizit\"at, 1995Link to Google Scholar