Robot arm problem

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
or in the COPS library.

Model formulation

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

The moving robot is modelled with the following equations:

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

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

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

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

Optimization problem

$\begin{array}{ll}\underset{x,u,t_f}{\min }& t_f\\ \text{s.t.}& \ddot{\rho }& =& \frac{u_{\rho }}{L},\\ & \ddot{\theta }& =& \frac{u_{\theta }}{I_{\theta }},\\ & \ddot{\mathrm{\Phi }}& =& \frac{u_{\mathrm{\Phi }}}{I_{\mathrm{\Phi }}},\\ & x(0)& =& (4.5,0,\frac{\pi }{4}{)}^T,\\ & x(t_f)& =& (4.5,\frac{2\pi }{3},\frac{\pi }{4}{)}^T,\\ & \dot{x}(0)& =& (0,0,0{)}^T,\\ & \dot{x}(t_f)& =& (0,0,0{)}^T,\\ & \rho (t)& \in & [0,L],\\ & \theta (t)& \in & [-\pi ,\pi ],\\ & \mathrm{\Phi }(t)& \in & [0,\pi ],\\ & u_{\rho }& \le & 1,\\ & u_{\theta }& \le & 1,\\ & u_{\mathrm{\Phi }}& \le & 1.\\ \end{array}$

where $I$ is the moment of inertia as above.

Source Code

Model descriptions are available in

• AMPL/TACO code at Robot arm problem (TACO)

== References ==

[Moessner1995]

M. Moessner-Beigel (1995): Optimale Steuerung f\"ur Industrieroboter unter Ber\"ucksichtigung der getriebebedingten Elastizit\"at. Ruprecht-Karls-Universit\"at Heidelberg