theory

You will model the motion of a lunar lander and then control the lander so it lands on a given target on the Moon. At first, you will neglect fuel loss. However, this is not realistic, so in later exercises you will include it. Our lunar lander has similar mass and thrust as the Apollo lander, but that’s where the similarities end.

The $- y$ direction is defined in the direction of the gravitational force by the Moon (downward). The lander has three thrusters that provide a constant thrust in the $+ x$ , $- x$ , and $+ y$ directions, respectively. Thrust is provided by the expulsion of gases (exhaust). Thus, the righward thrust, showing both the exhaust and the force vector, looks like:

force-right

The leftward thrust looks like:

force-left

A free-body diagram showing all thrusters engaged simultaneously along with the gravitational force by the Moon is shown below. However, we will assume for these exercises that thrusters cannot be engaged simultaneously. After completing the exercises, you are welcome to explore additional features of your model, such as allowing simultaneous thrusters for example.

force-all

At any instant, the net force on the rocket is

\begin{matrix} (1) & {\vec{F}}_{n e t} = {\vec{F}}_{g r a v} + {\vec{F}}_{t h r u s t} \end{matrix}

where ${\vec{F}}_{t h r u s t}$ can be any of the following:

$< 0, 0, 0 >$ if no engines are firing.
$F_{t h r u s t} < 1, 0, 0 >$ if a thruster exerts a force to the right.
$F_{t h r u s t} < - 1, 0, 0 >$ if a thruster exerts a force to the left.
$F_{t h r u s t} < 0, 1, 0 >$ if a thruster exerts a force upward.

The motion of the rocket is computed using the Momentum Principle:

\begin{matrix} (2) & \frac{d \vec{p}}{d t} = {\vec{F}}_{n e t} . \end{matrix}

For speeds much less than the speed of light:

\begin{matrix} (3) & \frac{d \vec{r}}{d t} \approx \frac{\vec{p}}{m} . \end{matrix}

Technically, the expulsion of gases means that the lander loses mass when a thruster is firing. Initially, let’s neglect mass loss due to burning fuel. However, in Exercises 3-5, we will include mass loss due to burning fuel. If including mass loss for the rocket (due to burning fuel), then in each time step:

\begin{matrix} (4) & m_{f} = m_{i} + \frac{d m}{d t} Δ t \end{matrix}

where $\frac{d m}{d t}$ is the fuel burn rate in kg/s and is negative. For the special case of a vertically descending or ascending rocket with thruster engaged, an analytic expression for the y-velocity of the rocket can be derived. An analytic calculation can be compared to the numeric calculation for this special case in order to test the computational model.