Moon Landing: Difference between revisions

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

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Latest revision as of 17:25, 22 February 2026

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

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

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 $t_{f}$ 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

@@ Line 14: / Line 14: @@
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{T} && -m(t_f) \\
+  \displaystyle \min_{T, t_\mathrm{f}} && -m(t_\mathrm{f}) \\
   \text{subject to} \\
 \quad \dot{h}(t) & = & v(t),\\
-\quad \dot{v}(t) & = & -1 + \frac{T(t)}{m}, \\
+\quad \dot{v}(t) & = & -1 + \frac{T(t)}{m(t)}, \\
 \quad \dot{m}(t) & = & -\frac{T(t)}{2.349}, \\
 \quad h(0) &=& 1, \\
 \quad v(0) &=& -0.783, \\
 \quad m(0) &=& 1, \\
-\quad t_f &\geq& 0, \\
+\quad t_\mathrm{f} &\geq& 0, \\
-\quad h(t_f) &=& 0, \\
+\quad h(t_\mathrm{f}) &=& 0, \\
-\quad v(t_f) &=& 0, \\
+\quad v(t_\mathrm{f}) &=& 0, \\
-\quad T(t) & \in & [0, 1.227] \ \quad \forall t \in [0,t_f]
+\quad T(t) & \in & [0, 1.227] \ \quad \forall t \in [0,t_\mathrm{f}]
    \end{array}
 </math>
@@ Line 35: / Line 35: @@
 <gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
-  Image:Moon_Landing.png| States and discretized control for a local optimum.
+  Image:Moon_Landing.png| States and discretized control for a local optimum. The free end time <math>t_\mathrm{f}</math> was modeled using the additional control <math>t</math>.
 </gallery>