Robot arm problem (TACO)

This page contains a model of the Robot arm problem in AMPL format, making use of the TACO toolkit for AMPL control optimization extensions. This problem is due to [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
. The original model using a collocation formulation can be found in the COPS library. Note that you will need to include a generic AMPL/TACO support file, OptimalControl.mod. To solve this model, you require an optimal control or NLP code that uses the TACO toolkit to support the AMPL optimal control extensions.

AMPL

This is the source file robotarm_taco.mod

# ----------------------------------------------------------------
# Robot arm problem using AMPL and TACO
# (c) Christian Kirches, Sven Leyffer
#
# Source: COPS 3.1 collocation formulation - March 2004
#         David Bortz - Summer 1998          
# ----------------------------------------------------------------
include OptimalControl.mod;
 
# Final positions of the length and the angles for the robot arm
 
param L;		# total length of arm
param pi := 4*atan(1);
 
# Upper bounds on the controls
 
param max_u_rho;
param max_u_the;
param max_u_phi;
 
# Initial positions of the length and the angles for the robot arm
 
param rho_0;
param the_0;
param phi_0;
 
# Final positions of the length and the angles for the robot arm
 
param rho_n;
param the_n;
param phi_n;
 
# The length and the angles theta and phi for the robot arm.
 
var rho >=0, <= L; 
var the >= -pi, <= pi; 
var phi >=0, <= pi;
 
# The derivatives of the length and the angles.
 
var rho_dot;
var the_dot;
var phi_dot;
 
# The controls.
 
var u_rho >= -max_u_rho, <= max_u_rho, suffix type "u0";
var u_the >= -max_u_the, <= max_u_the, suffix type "u0";
var u_phi >= -max_u_phi, <= max_u_phi, suffix type "u0";
 
# The independent time and the final time.
 
var t;
var tf := 1.0, >= 0.1, <= 12 suffix scale 10.0;
 
# The moments of inertia.
 
var I_the = ((L-rho)^3+rho^3)*(sin(phi))^2/3.0;
var I_phi = ((L-rho)^3+rho^3)/3.0;
 
# The robot arm problem.
 
minimize time: eval(t,tf) suffix scale 1e+2;
 
subject to rho_eqn:
        diff(rho, t) = rho_dot;
 
subject to the_eqn:
        diff(the, t) = the_dot;
 
subject to phi_eqn:
        diff (phi, t) = phi_dot;
 
subject to u_rho_eqn:
        diff (rho_dot, t) = u_rho/L;
 
subject to u_the_eqn:
        diff (the_dot, t) = u_the/I_the;
 
subject to u_phi_eqn:
        diff (phi_dot, t) = u_phi/I_phi;
 
# Boundary Conditions
 
subject to rho_0_eqn: eval (rho, 0) = 4.5 suffix type "dpc";
subject to the_0_eqn: eval (the, 0) = 0 suffix type "dpc";
subject to phi_0_eqn: eval (phi, 0) = pi/4 suffix type "dpc";
 
subject to rho_f_eqn: eval (rho, tf) = 4.5;
subject to the_f_eqn: eval (the, tf) = 2*pi/3;
subject to phi_f_eqn: eval (phi, tf) = pi/4;
 
subject to rho_dot_0_eqn: eval (rho_dot, 0) = 0 suffix type "dpc";
subject to the_dot_0_eqn: eval (the_dot, 0) = 0 suffix type "dpc";
subject to phi_dot_0_eqn: eval (phi_dot, 0) = 0 suffix type "dpc";
 
subject to rho_dot_f_eqn: eval (rho_dot, tf) = 0;
subject to the_dot_f_eqn: eval (the_dot, tf) = 0;
subject to phi_dot_f_eqn: eval (phi_dot, tf) = 0;
 
option solver ...;
 
solve;

Other Descriptions

Other descriptions of this problem are available in

• Mathematical notation at Robot arm problem
• AMPL (using a fixed discretization) at the COPS library

References

[Moessner1995]

M. Moessner-Beigel: Optimale Steuerung f\"ur Industrieroboter unter Ber\"ucksichtigung der getriebebedingten Elastizit\"at, 1995