We consider the problem of landing a spacecraft on the Moon, assuming that aerodynamic and gravitational forces of bodies other than the Moon are negligible, and lateral motion can be ignored. Accordingly, the descent trajectory is vertical, and the thrust vector is tangent to the trajectory.
Because the spacecraft is near the Moon, we assume that the lunar acceleration of gravity has the constant value , that the relative velocity of the exhaust gases with respect to the spacecraft is constant, and that the mass rate is constrained by , where is constant and gives the maximum rate of change of the mass due to burning the fuel.
Mathematical Approach to a Soft Landing
The problem is to make a soft landing on the surface of the Moon with the minimum amount of fuel.
Here is a sketch of the system immediately preceding the landing.
- is time
- is the mass of the spacecraft, which varies as fuel is burned
- is the rate of change of mass, constrained by
- , the gravitational constant near the Moon
- is a constant, the relative velocity of the exhaust gases with respect to the spacecraft
- , the thrust
- is the the height, with
- , the velocity
- , the control function
Recalling our assumptions, aerodynamic forces and gravitational forces of bodies other than the Moon are negligible and lateral motion is ignored. Thus the descent trajectory is vertical and the thrust vector is perpendicular to the ground.
We also suppose that , where is the mass of the spacecraft without fuel and is the initial mass of fuel; , since as we expect that the spacecraft will return to Earth, it needs some fuel for takeoff.
Equations of Motion
which can be written as a system of equations
where is a constant. The third equation states that the loss of mass per second (the fuel burned by the jet per second) is proportional to the thrust of the jet.
The Optimal Control Problem
Our goal is to minimize the fuel consumption, so the cost functional is
where is the first time for which
Thus the horizon is , where remains to be determined.
In vector form, if , then
and the problem can be written
From (3) we have that
and by integration over the interval , we get that
It follows that if and only if
Solving for ,
Now we substitute this into (5) to get
This result was published in .
From here it follows that instead of (7) we can consider the following cost functional
Necessary Conditions for the Mayer Problem
To avoid a lengthy discussion, we state a short version of theorem 4.2.i in . Let the Mayer problem be expressed as
A pair , , is said to be admissible or feasible provided is absolutely continuous , is measurable, and and satisfy (10). Let be the class of admissible pairs . The goal is to find the minimum of the cost functional over , that is, to find an element such that for all . We introduce the variables , called multipliers, and an auxiliary function , called the Hamiltonian, defined on by
More assumptions are necessary:
- There exists an element such that for all .
- is closed in .
- The set , is closed in .
- The graph of the optimal trajectory belongs to the interior of .
- does not depend on time and is a closed set.
- The end point of the optimal trajectory is a point of , where possesses a tangent variety (of some dimension , ), whose vectors are denoted by
so that , , , where and are constants.
For , the minimum of is attained with , and then
This corresponds to free fall for the spacecraft.
For , the minimum of is attained with , and then
Thus we find that the control function takes only extreme values and .
If on an interval we have that , and hence
then for , we have
In this case, describes an arc of a parabola of equation
If on an interval we have , and hence
then for , we find
Program for Soft Landing on the Moon
MoonLanding is a Mathematica program for a soft landing on the Moon. Here h0 is the initial height, v0 is the initial velocity, mass is the mass of the lander without fuel, fuel is the initial fuel, g is acceleration due to gravity, k is the relative velocity of the exhaust gases, and is the rate of change of the mass by burning.
The correctness of the results drastically depends on the initial values of the variables z and g that we use in solving the nonlinear system of equations in the program.
This Manipulate lets you vary the parameters in real time.
The author expresses his gratitude to Horia F. Pop from Babes-Bolyai University, Faculty of Mathematics and Computer Science in Cluj-Napoca, Romania, for valuable discussions.
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|Marian Mureşan, “Soft Landing on the Moon with Mathematica,” The Mathematica Journal, 2012. dx.doi.org/doi:10.3888/tmj.14-13.|
About the Author
Marian Mureşan is affiliated with Babes-Bolyai University, Faculty of Mathematics and Computer Science, in Cluj-Napoca, Romania. He is interested in analysis, nonsmooth analysis, calculus of variations, and optimal control.