Example--Airplane Takeoff :: Demo: Vector Transition Processing Using Windows (Signal Processing Toolbox)

Signal Processing Toolbox

Example--Airplane Takeoff

Consider the problem of designing a smooth takeoff path for an airplane. The initial vector is the plane heading down the runway just before takeoff. The final vector is the level flight path at a given altitude. You must design the path the plane takes between these two vectors while avoiding jarring the passengers too much during the takeoff. The two key design goals for this problem are:

A smooth transition path
No discontinuities (in position, velocity or acceleration)

To view this example, start the Vector Window demo

```
windtrandemo
```

Set the following options

Window Selection
Hann

First-Order Method
Mirror

Sample Vectors
Raster Y

Second Derivative Plot
Acceleration

`Window Selection`	`Hann`
`First-Order Method`	`Mirror`
`Sample Vectors`	`Raster Y`
`Second Derivative Plot`	`Acceleration`

The Vector Transition plot shows one possible smoothed trajectory (in blue) compared to the direct path (in red). The blue trace seems sufficiently gradual to make a good choice.

How was this path computed? Look at the Second Derivative (Acceleration) plot, which has two back-to-back Hann windows (hann) . You could create a similar plot with

```
plot([hann(64); -1*hann(64)])
```

The Hann window was used to define the acceleration during the vector transitions. The actual trajectory waveform was created by integrating the acceleration curve twice, with appropriate initial conditions.

A plot of position during the takeoff period can be created with

plot(cumsum(cumsum([hann(64); -1*hann(64)])));

A plot of velocity during the takeoff period can be created with

```
plot(cumsum([hann(64); -1*hann(64)]));
```

Try changing the window selection type. You will see different trajectories, with different peak accelerations and spectral characteristics. Select Energy Spectra in Second Derivative Plot to see the spectra, which are closely tied to the underlying window function. Some windows are better than others at creating a smooth path between vectors. (Wouldn't you prefer taking off in a plane that uses a Hann window (hann) rather than a Chebyshev (chebwin) with parameter = 1?)

Demo: Vector Transition Processing Using Windows Vector Transition Demo