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yuri_base

How to calculate true airspeed on a plane jump

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Yo!

So you bought your 20lbs. "Fly Like A Pro" clown shoes - leather tops with replaceable lead soles - and are eager to let them fly your body in the clouds forever.

How do you calculate L/D on a plane jump? How to eliminate wind from your GPS data? Here's the method.

Your groundspeed G (bold font denotes vector, regular denotes vector's length) is equal to the vector sum of your horizontal airspeed A and windspeed W:

G = A + W

Your GPS data shows G, but how do you extract A from this sum if you don't know W?

A very crude approach is to use the winds aloft forecast for W. However, the forecast is usually far from being accurate for our purposes.

Another approach is to fly directly into the wind on one leg of your flight, then turn 180 and fly downwind. The average of the two groundspeeds will give you your airspeed, since the wind will be cancelled. But again, you need to know the direction of the wind, and this direction is oftentimes incompatible with the jump plan.

Can we eliminate the wind while flying an arbitrary pattern?

Suppose you fly at a constant speed A and make two legs at some angle to each other.

On the 1st leg,

G1 = A1 + W,

and on the 2nd leg,

G2 = A2 + W.

By subtracting the two equations, we eliminate the wind:

G1 - G2 = A1 - A2.

Airspeed vector A can be represented as a unit vector B (B = 1) multiplied by A: A = A*B. B is your magnetic bearing vector.

Thus,

A*(B1 - B2) = G1 - G2

and

A = |G1 - G2|/|B1 - B2|

(|...| denotes vector's length)

If g is the course of your groundspeed and b is magnetic bearing (both measured in degrees), then

|G1 - G2| = sqrt(G1^2 + G2^2 - 2*G1*G2*cos(g1-g2))

|B1 - B2| = sqrt(B1^2 + B2^2 - 2*B1*B2*cos(b1-b2)) = sqrt(1 + 1 - 2*1*1*cos(b1-b2))

and finally

A = sqrt((G1^2 + G2^2 - 2*G1*G2*cos(g1-g2))/(2 - 2*cos(b1-b2)))

For two particular cases - when you change your bearing by 90 or 180 degrees, the formula simplifies to:

for b1 - b2 = 90
A = sqrt((G1^2 + G2^2 - 2*G1*G2*cos(g1-g2))/2)

for b1 - b2 = 180
A = sqrt(G1^2 + G2^2 - 2*G1*G2*cos(g1-g2))/2

So here's what you do.

1. Put your magic clown shoes on. ;)

2. Find two reference points far on horizon at 90 or 180 degrees to each other - mountains, lakes, etc. They should be very far so they set the direction (bearing) for your flight, not the course on the ground. Sun is not very practical, since you need to lift your head up to check your bearing and that changes your speed. Another possibility is to use compass mounted on your chest strap.

3. Fly two legs at the same speed maintaining your heading/bearing in the chosen directions.

4. Plug the averaged groundspeeds G1 and G2 and the courses g1 and g2 into the formula above. Calculate your true L/D.

5. Rebalance your shoes and repeat. :)
Of course, windspeed varies with height, so you want to make both legs within the layer of more-or-less constant speed. Winds aloft forecast will give you an idea if the winds are too 'layered' or smooth.

Yuri
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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forgive me if i'm missing something, but i'm curious...

If you are flying at two different angles on two different legs, then you are flying at two different angles relative to the wind as well, right? so:

G1 = A + (some fraction of) W and
G2 = A + (some other fraction of) W

W can't just cancel out unless they are the same. To do that, you'd still need to know the wind's direction and plan the jump accordingly.

Disclaimer: I didn't have time to look at all the equations you mentioned, and maybe you somehow accounted for this (maybe thats what the trig functions are in there for), but it just doesn't seem quite right to me. Please elaborate :)
-Ghetto
"The reason death cannot frighten me, is because life has cured me of fear."
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Cleveland Skydiving

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Or you can use Paralogs new wind correction option that does all of that and more for you. Stay tuned for more groovy wind calculation information once this version of the software is released.;)

Attached is a screenshot of the new GPS Import dialog where you can enter wind data. A
similar screen will be available for the jump profile viewer so that you can set the data for a profile later.

The wind data entered in these tables will be interpolated and used to modify the horizontal components of the jump profile accordingly.
"It's just skydiving..additional drama is not required"
Some people dream about flying, I live my dream
SKYMONKEY PUBLISHING

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If you are flying at two different angles on two different legs, then you are flying at two different angles relative to the wind as well, right? so:

G1 = A + (some fraction of) W and
G2 = A + (some other fraction of) W



The trick is to use vector algebra, not scalar. Then you can cancel out the wind vector, so its actual direction and speed don't matter.
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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Or you can use Paralogs new wind correction option that does all of that and more for you. Stay tuned for more groovy wind calculation information once this version of the software is released.



In the method above, you sample the wind at place and time of your jump, instead of relying on the forecast numbers.
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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In the method above, you sample the wind at place and time of your jump, instead of relying on the forecast numbers.



But this assumes that a) the wind and b) your performance is constant during both legs while both usually change during your flight which will spoil this calculation.

I agree that forecasted data is not the best solution, but I am afraid this is the best we can do. Ideal solution would be to drop a GPS on a streamer just before exit, but you better combine it with a GSM transmitter so you can find it again.

I read about pilots use a similar technique by flying a triangle at the same level and using GPS data only to calculate the true wind.
My Logbook

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If you are flying at two different angles on two different legs, then you are flying at two different angles relative to the wind as well, right? so:

G1 = A + (some fraction of) W and
G2 = A + (some other fraction of) W



The trick is to use vector algebra, not scalar. Then you can cancel out the wind vector, so its actual direction and speed don't matter.



Whatever dude....

You have made some very impractical assumptions in the simplification to assume that the wind cancels out.

Kris.

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Bring 2 GPS-es on a plane jump.
Drop one out of the door, wear the other one.

After your jump, go on a long walk, and hope to find a working GPS embedded in the dirt somewhere..
Then check the data and you're there! You know the winds at altitude!
JC
FlyLikeBrick
I'm an Athlete?

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Here is the error analysis for non-uniform winds.

If the windspeed on the 1st leg is W1 and on the 2nd W2,

G1 = A1 + W1,
G2 = A2 + W2,

so

A = |G1 - G2 + dW|/|B1 - B2|

where dW = W2 - W1 is the [vector] differential of windspeed between the layers.

For "U-turn" pattern, |B1 - B2| = 2, so the maximum error we get in A is

dA = |dW|/2

For 90-degree pattern,

dA = |dW|/1.4

For example, if the wind on the 1st leg is 30mph average and on the 2nd leg 20mph average, with change in direction 30 degrees, then |dW| = sqrt(30*30 + 20*20 - 2*30*20*cos(30)) = 16mph, so for U-turn the error in calculated airspeed is 8mph, for 90-turn 11mph. If your fallrate is 35mph, this translates to an error in L/D of 0.2 and 0.3, respectively. Quite an error, but it's still better than no wind correction at all.

Yuri
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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Yo!

Klaus of Paralog digged out the information about a very clever way to calculate true airspeed (TAS) using 3-leg pattern. This approach doesn't require to fly the legs in predetermined directions like above.

See "Measuring True Airspeed using GPS" at the end of this page:
http://go.phpwebhosting.com/~khorton/rv8/phplinks/index.php?&PID=49

(I attached the article and the spreadsheet from this link for convenience.)

It was developed by Doug Gray in 1998. Very elegant geometric solution!!! (gives a mental erection to those who like math) ;)

Here's what to do.

1. Fly a 3-leg pattern trying to maintain the same airspeed as best as you can. Freeze your body position and make turns in such a way that you can restore exactly the same body position. Fly straight legs long enough to collect enough datapoints but short enough to stay in the layer of the same windspeed (check the winds aloft forecast to see if the wind from 6K to 12K is about constant both in speed and direction). An equilatteral triangle pattern will give the most accurate result, but legs at 90 degrees will work great, too.

2. Calculate the average groundspeed and course for each leg from your GPS data.

3. Plug the numbers into Gray's spreadsheet and you're done!

Many thanks to Klaus for productive discussion and finding this great stuff.

Woohoo! :)
Yuri
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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On the second thought, the 2-leg 180-degree method described in the first post probably beats the 3-leg method by simplicity while having very good accuracy.

The 3-leg method is based on the fact that 3 points (ends of groundspeed vectors drawn from a common point) define a circle (the radius of which is TAS).

The 2-leg method is based on the fact that 2 points also define a circle if we know the angle between radii for these points (in other words, if we know the difference in TAS bearing). In particular, the most accurate case, the 2 points are the ends of the diameter (2 legs with 180-degree change in bearing).

How sensitive is the 2-leg method to the accuracy of the U-turn? A simple calculation shows that the error db in bearing of the 2nd leg results in relative error dA/A of TAS equal to

dA/A = 1 - cos(db/2)

For example, if you made a U-turn and your new bearing (relative to the 1st one) is 150 degrees instead of 180, TAS if off by

dA/A = 1 - cos(30/2) = 0.03 = 3%

That's typically just 3mph error - pretty damn good for a quite sloppy U-turn.

The advantage of the 2-leg method is that it consumes less altitude - less chance of entering a layer with different windspeed - and only one turn is needed - less chance of distorting the body position and flying legs at different airspeeds.

To sum it up:

1. Fly parallel to some straight feature on the ground (road). Make U-turn, fly parallel to the road again. ("fly parallel" means keep your body oriented parallel to road; the wind will actually blow you sideways (usually not noticeable, especially at altitude), and your actual path across the ground will be at some angle to the road).

2. Your true airspeed is:

TAS = sqrt(G1^2 + G2^2 - 2*G1*G2*cos(g1-g2))/2

Yuri
Android+Wear/iOS/Windows apps:
L/D Vario, Smart Altimeter, Rockdrop Pro, Wingsuit FAP
iOS only: L/D Magic
Windows only: WS Studio

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The math is interesting but it does not give me an erection anymore :P.

FWIW.

Learning to fly in as close to a straightline as possible is more important for getting max glide than anything else.

Learning to fly with total concentration and without as much as a twitch is what will put one on the path to learning to fly at their best glide. Looking at a road or the tandem that left before you or anything else will just make the process of learning slower.

In the end is all this calculating not about learning to fly efficiently?

Kris.

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