rocket.gms : Goddard rocket COPS 2.0 #10

**Description**

Maximize the final altitude of a vertically launched rocket, using the thrust as a control and given the initial mass, the fuel mass, and the drag characteristics of the rocket. This model is from the COPS benchmarking suite. See http://www-unix.mcs.anl.gov/~more/cops/. The number of discretization points can be specified using the command line parameter. COPS performance tests have been reported for nh = 50, 100, 200, 400

**References**

- Dolan, E D, and More, J J, Benchmarking Optimization Software with COPS. Tech. rep., Mathematics and Computer Science Division, 2000.
- Bryson, A E, Dynamic Optimization. Addison Wesley, 1999.

**Large Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** rocket.gms

```
$title Goddard Rocket COPS 2.0 #10 (ROCKET,SEQ=238)
$onText
Maximize the final altitude of a vertically launched rocket, using the
thrust as a control and given the initial mass, the fuel mass, and the
drag characteristics of the rocket.
This model is from the COPS benchmarking suite.
See http://www-unix.mcs.anl.gov/~more/cops/.
The number of discretization points can be specified using the command
line parameter. COPS performance tests have been reported for nh = 50,
100, 200, 400
Dolan, E D, and More, J J, Benchmarking Optimization
Software with COPS. Tech. rep., Mathematics and Computer
Science Division, 2000.
Bryson, A E, Dynamic Optimization. Addison Wesley, 1999.
Keywords: nonlinear programming, aerospace engineering, Goddard rocket
$offText
$if not set nh $set nh 50
Set h 'intervals' / h0*h%nh% /;
Scalar
h_0 'initial height' / 1 /
v_0 'initial velocity' / 0 /
m_0 'initial mass' / 1 /
g_0 'gravity at the surface' / 1 /
nh 'number of intervals in mesh' / %nh% /
t_c 'thrust constant' / 3.5 /
v_c / 620 /
h_c / 500 /
m_c / 0.6 /
D_c
m_f 'final mass'
c;
Variable final_velocity;
Positive Variable
step 'step size'
v(h) 'velocity'
ht(h) 'height'
g(h) 'gravity'
m(h) 'mass'
t(h) 'thrust'
d(h) 'drag';
Equation
df(h) 'drag function'
gf(h) 'gravity function'
obj
h_eqn(h)
v_eqn(h)
m_eqn(h);
obj.. final_velocity =e= ht('h%nh%');
df(h).. d(h) =e= D_c*sqr(v(h))*exp(-h_c*(ht(h) - h_0)/h_0);
gf(h).. g(h) =e= g_0*sqr(h_0/ht(h));
h_eqn(h-1).. ht(h) =e= ht(h-1) + .5*step*(v(h) + v(h-1));
m_eqn(h-1).. m(h) =e= m(h-1) - .5*step*(T(h) + T(h-1))/c;
v_eqn(h-1).. v(h) =e= v(h-1) + .5*step*((T(h) - D(h) - m(h)*g(h))/m(h)
+ (T(h-1) - D(h-1) - m(h-1)*g(h-1))/m(h-1));
c = 0.5*sqrt(g_0*h_0);
m_f = m_c*m_0;
D_c = 0.5*v_c*(m_0/g_0);
ht.lo(h) = h_0;
t.lo(h) = 0.0;
t.up(h) = T_c*(m_0*g_0);
m.lo(h) = m_f;
m.up(h) = m_0;
ht.fx('h0') = h_0;
v.fx('h0') = v_0;
m.fx('h0') = m_0;
m.fx('h%nh%') = m_f;
ht.l(h) = 1;
v.l(h) = ((ord(h) - 1)/nh)*(1 - ((ord(h) - 1)/nh));
m.l(h) = (m_f - m_0)*((ord(h) - 1)/nh) + m_0;
t.l(h) = t.up(h)/2;
step.l = 1/nh;
* Initial values for intermediate variables
d.l(h) = D_c*sqr(v.l(h))*exp(-h_c*(ht.l(h) - h_0)/h_0);
g.l(h) = g_0*sqr(h_0/ht.l(h));
Model rocket / all /;
$if set workSpace rocket.workSpace = %workSpace%
solve rocket using nlp maximizing final_velocity;
```