Math Help - curve fitting of real time flight position data

1. curve fitting of real time flight position data

Hi All,

I have got some real time position data of a flight. I need to get a curve that best fits these data. How do I go about?

Srini

2. Start by drawing a scatterplot, so that you get an idea of the "type" of function you will be dealing with (Linear, Quadratic, Exponential etc...)

Then apply that kind of regression.

3. Originally Posted by sriecewit
Hi All,

I have got some real time position data of a flight. I need to get a curve that best fits these data. How do I go about?

Srini
You need to provide more information, what exactly is the data, what are its error charateristics, what was the platforms nominal motion, ...

... and of course what you want to do with the result.

CB

4. Originally Posted by CaptainBlack
You need to provide more information, what exactly is the data, what are its error charateristics, what was the platforms nominal motion, ...

... and of course what you want to do with the result.

CB

I have the 3 dimensional position of a flight in x, y, and z co-ordinates (ECEF they call it) in cartesian co-ordinate system. i.e. if the flights position 'P' is designated by these three co-ordinates, P(x,y,z), I have got this position at different instants of time at an interval of 100ms.

Sample data is as follows:
-1138227.9178 -5370424.0963 3257208.3655
-1138240.1536 -5370414.0210 3257220.6192
-1138252.3895 -5370403.9456 3257232.8728
-1138264.6253 -5370393.8702 3257245.1265
-1138276.8611 -5370383.7947 3257257.3801
-1138289.0968 -5370373.7191 3257269.6337
-1138301.3325 -5370363.6435 3257281.8873
-1138313.5682 -5370353.5679 3257294.1409
-1138325.8038 -5370343.4921 3257306.3945
-1138338.0395 -5370333.4164 3257318.6480
-1138350.2750 -5370323.3405 3257330.9015
-1138362.5106 -5370313.2646 3257343.1551
-1138374.7461 -5370303.1887 3257355.4086
-1138386.9816 -5370293.1126 3257367.6620
-1138399.2171 -5370283.0366 3257379.9155
-1138411.4525 -5370272.9604 3257392.1690
-1138423.6879 -5370262.8842 3257404.4224
-1138435.9233 -5370252.8080 3257416.6758
-1138448.1586 -5370242.7317 3257428.9292
-1138460.3939 -5370232.6553 3257441.1826
-1138472.6292 -5370222.5789 3257453.4359
-1138484.8644 -5370212.5024 3257465.6893
-1138497.0996 -5370202.4259 3257477.9426
-1138509.3348 -5370192.3493 3257490.1959
-1138521.5700 -5370182.2726 3257502.4492
-1138533.8051 -5370172.1959 3257514.7025
-1138546.0402 -5370162.1191 3257526.9558
-1138558.2752 -5370152.0423 3257539.2090
-1138570.5102 -5370141.9654 3257551.4623
-1138582.7452 -5370131.8885 3257563.7155
...
and so on.

I need to find a mathematical model of the flight's position which fits this data.

So that I can best replicate the position at any time.

5. Originally Posted by sriecewit
I have the 3 dimensional position of a flight in x, y, and z co-ordinates (ECEF they call it) in cartesian co-ordinate system. i.e. if the flights position 'P' is designated by these three co-ordinates, P(x,y,z), I have got this position at different instants of time at an interval of 100ms.

Sample data is as follows:
-1138227.9178 -5370424.0963 3257208.3655
-1138240.1536 -5370414.0210 3257220.6192
-1138252.3895 -5370403.9456 3257232.8728
-1138264.6253 -5370393.8702 3257245.1265
-1138276.8611 -5370383.7947 3257257.3801
-1138289.0968 -5370373.7191 3257269.6337
-1138301.3325 -5370363.6435 3257281.8873
-1138313.5682 -5370353.5679 3257294.1409
-1138325.8038 -5370343.4921 3257306.3945
-1138338.0395 -5370333.4164 3257318.6480
-1138350.2750 -5370323.3405 3257330.9015
-1138362.5106 -5370313.2646 3257343.1551
-1138374.7461 -5370303.1887 3257355.4086
-1138386.9816 -5370293.1126 3257367.6620
-1138399.2171 -5370283.0366 3257379.9155
-1138411.4525 -5370272.9604 3257392.1690
-1138423.6879 -5370262.8842 3257404.4224
-1138435.9233 -5370252.8080 3257416.6758
-1138448.1586 -5370242.7317 3257428.9292
-1138460.3939 -5370232.6553 3257441.1826
-1138472.6292 -5370222.5789 3257453.4359
-1138484.8644 -5370212.5024 3257465.6893
-1138497.0996 -5370202.4259 3257477.9426
-1138509.3348 -5370192.3493 3257490.1959
-1138521.5700 -5370182.2726 3257502.4492
-1138533.8051 -5370172.1959 3257514.7025
-1138546.0402 -5370162.1191 3257526.9558
-1138558.2752 -5370152.0423 3257539.2090
-1138570.5102 -5370141.9654 3257551.4623
-1138582.7452 -5370131.8885 3257563.7155
...
and so on.

I need to find a mathematical model of the flight's position which fits this data.

So that I can best replicate the position at any time.
Try plotting the data (I will give you a clue, the data given looks like a line in 3D)

Each point is a constant increment on the last.

CB

6. But do you know that the subsequent data will be in a straight line?

There is time information, at least relative time information. You start the clock at t=0 at the first data point, and we're told that the sampling rate is 10 Hz.

7. First you should decide should your model interpolate the given data or not. Then you should decide on how to measure the goodness of fit. Absolute deviation, least square, etc...

In case you don't want to go that way here's what I've noticed.
Simply by looking at the data you provided I noticed the following:

- the gap between the numbers in the first column is roughly 12 (step lenght -12)
- the gap between the numbers in the first column is roughly 10 (step lenght +10)
- the gap between the numbers in the first column is roughly 12 (step lenght +12)

This could lead to a parametric model, maybe something like this:
$\begin{array}{l}x(t)=x_0+t\cdot \Delta x,\\ y(t)=y_0+t\cdot \Delta y,\\ z(t)=z_0+t\cdot \Delta z,\end{array}$
where $\Delta x,\,\Delta y,\,\Delta z$ are mentioned step lengths. You could estimate them using the average value of differences between every two consecutive values, i.e. using the "rough values"

$\begin{array}{l}x(t)=x_0 - 12t,\\ y(t)=y_0+10t,\\ z(t)=z_0+12t,\end{array}$

Naturally, I'm just giving you one empirical approach. Curve fitting is much more complicated, even when you have 2D data, let alone 3D.