Goddart's rocket problem (TACO)

This page contains a model of the Goddart's rocket problem in AMPL format, making use of the TACO toolkit for AMPL control optimization extensions. 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 goddart_taco.mod

# ----------------------------------------------------------------
# Goddart's rocket problem using AMPL and TACO
# (c) Christian Kirches, Sven Leyffer
#
# Source: COPS 3.1 collocation formulation - March 2004
# ----------------------------------------------------------------
include OptimalControl.mod;
 
var t;
 
var tf := 1.0, >= 0.01, <= 1;
 
param v0 := 0.0;
param m0 := 1.0;
param g0 := 1.0;
param h0 := 1.0;
param hc := 500.0;
param vc := 620.0;
param mc := 0.6;
param Tmax := 3.5*g0*m0;
param c := 0.5*sqrt(g0*h0);
param mf := mc*m0;
 
var h >= h0, := 1;
let h.interp_to := h0;
var v >= v0;
let v.interp_to := v0;
var m <= m0;
let m.interp_to := mf;
 
var T >= 0, <= Tmax, := Tmax/2;
let T.type := "u0";
 
param Dc := 0.5*vc*m0/g0;
var D = Dc*v^2*exp(-hc*(h-h0)/h0);
var g = g0*(h0/h)^2;
 
maximize FinalHeight: eval(h,tf);
let FinalHeight.scale := 0.01;
subject to
 
dh: diff(h,t) = v;
dv: diff(v,t) = (T - D)/m - g;
dm: diff(m,t) = -T/c;
 
ih: eval(h,0) = h0;
iv: eval(v,0) = v0;
im: eval(m,0) = m0;
fm: eval(m,tf) = mf;
 
option solver ...;
 
solve;

Other Descriptions

Other descriptions of this problem are available in

• Mathematical notation at Goddart's rocket problem
• AMPL (using a fixed discretization) at the COPS library