# Optimum Re-entry Corridor for Apollo-type Spacecraft

1. Apr 9, 2013

### roldy

I'm working on a project in MATLAB for my own personal benefit. I found a 1966 paper where the limits of the re-entry corridor are found. The control variable is the attack angle. The state variables are flight path angle, velocity, altitude, angular displacement (range), and penalty functions for altitude and g-force loading on the pilot. As of now, trying to work the limits of the re-entry corridor. This is done by numerically integrating the equations of motion and checking to see if the total g-force loading on the pilot is above a threshold. The penalty function for altitude is there in case the spacecraft skips out above the reference altitude of 150 km. I'm not concerned with this part. The pilot penalty function was digitized and an equation was fit to the curve. This equation is in the numerical integration function.

Note: I tried two different sets of equations for the equations of motion. It seems that the second set is working better and I'm getting reasonable re-entry time values.

For the time being, I'm just testing out the program (without optimizing the reentry flight path angle) with a flight path angle of 90 degrees. To check if this part of the program is working, I plotted Altitude vs. range. The range they have goes from 0 to 1. My values are on the order of 10-5.

The sets of equations I am using are as follows:
$\dot{V} = -D/m+ g\sin(\theta)$
$\dot{\phi} = -V\cos(\theta)/(R_E + alt)$
$\dot{\theta} = (L/m - g\cos(\theta))/V$
$\dot{alt} = -V\sin(\theta)$
$\dot{Pilot_{pen}} = 1/\tau$
The function $\tau$ is in the code.

$g = g_{surf}(R_E/(R_E + alt))^2$
$\rho = 1.752e^{-alt/6700}$

$C_d = 2(\sin(\alpha))^3$
$C_l = 2(\sin(\alpha))^2\cos(\alpha)$

$D = 1/2C_d\rho V^2S_{ref}$
$L = 1/2C_l\rho V^2S_{ref}$

$Aero_{accel} = \sqrt{(L^2 + D^2)/(mg_{surf})}$

Could someone take a look at my code and see if I am doing anything wrong? Here is the link for the zip file containing the code and the paper I am working from.

Last edited: Apr 9, 2013