## Space

### Emergency landing of a hypersonic flight system

This is an example of an emergency landing of a two stage space transport vehicle. After the separation of the first stage the engine of the upper stage cannot be re-ignited. Due to this propulsion system damage the upper stage cannot reach a safe orbit.

The optimal control problem is fully discretized using Euler's method and implemented in AMPL;
this leads to a sparse large-scale nonlinear optimisation problem. The problem dimensions are
determined by the number of points n used for the discretization, specifically it has
`N = 8n+1` optimisation variables subject to `M = 6(n-1)+4n+7` constraints.

The picture on the right shows an example of an emergency trajectory. The reentry point is situated over the city of Bremen, Germany and the objective function is to maximize the flight distance without propulsion from this reentry point.

The following table summarizes the results for different number of discrete points. The tests
were run on an Intel Core2 Quad CPU Q6600 @ 2.40GHz with tolerances for constraint violations of
10^{-6} and for KKT conditions of 10^{-5}.

n | N | M | Time |
---|---|---|---|

201 | 1609 | 2011 | 4.88s |

2001 | 16009 | 20011 | 45.47s |

5001 | 40009 | 50011 | 281.30s |

10001 | 80009 | 100011 | 424.25s |

20001 | 160009 | 200011 | 2261.35s |

40001 | 320009 | 400011 | 4898.53s |

### Ascent trajectories

WORHP was succesfully integrated within the ESA project "eNLP" in the Astos software. Within the software it is possible to describe ascent optimal control problems for rockets, e.g. Ariane V.

**Ascent trajectory for an Ariane V from Kourou within Astos 6.1**