Moon Landing

Moon Landing
State dimension:	1
Differential states:	3
Discrete control functions:	2

The Moon Landing problem is a simplification of a spacecraft trying to land on the moon's surface. Its objective is to minimize the fuel consumption during the landing maneuver while landing savely on the ground with zero vertical velocity.

The implementation here is taken from [1]. Its dynamics are given by a two-dimensional ODE model.

Mathematical formulation

$\begin{array}{lll}\underset{T}{\min }& & -m(t_{\mathrm{f}})\\ \text{subject to}\\ \dot{h}(t)& =& v(t),\\ \dot{v}(t)& =& -1+\frac{T(t)}{m},\\ \dot{m}(t)& =& -\frac{T(t)}{2.349},\\ h(0)& =& 1,\\ v(0)& =& -0.783,\\ m(0)& =& 1,\\ t_{\mathrm{f}}& \ge & 0,\\ h(t_{\mathrm{f}})& =& 0,\\ v(t_{\mathrm{f}})& =& 0,\\ T(t)& \in & [0,1.227]\mathrm{\forall }t\in [0,t_{\mathrm{f}}]\end{array}$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum. The free end time $t_{\mathrm{f}}$ was modeled using the additional control $t$.

Miscellaneous and Further Reading

This formulation and a detailed description can be found in [1].

References

[1] Multidisciplinary Optimal Control Library: https://openmdao.org/dymos/docs/latest/examples/moon_landing/moon_landing.html