Calculation & Worksheets

Using GPS to accurately establish True Airspeed (TAS).

June, 1998
Doug Gray
5 Larkspur Place, Heathcote, NSW Australia 2233
Phone/Fax 61 2 9520 5542
Email: dgra1233@bigpond.net.au
Background
True Airspeed or TAS and its determination is an essential part in the assessment of aircraft
performance. From airspeed indicator calibration to performance evaluation it is necessary to
know or determine TAS accurately.
The following describes yet another method for determining TAS but in this case it allows a
little more flexibility in flying the aircraft. Cramped airspaces should no longer be a limiting
factor.
Using the GPS
One very good technique1 described by David Fox requires the aircraft to be flown
accurately over specific ground tracks, with the recorded speeds combined in a simple
computation to derive the TAS. In fact this is a special case of the general solution to the
problem.
The general solution still requires three legs but these can be (almost) any heading/track.
The aviating will be more convenient, but the calculation will be more difficult, hence I
recommend a spreadsheet. After noting down the GPS Speed plus the GPS Track for each leg,
these six figures are entered to arrive at the TAS.
The following illustration is helpful to understand the geometry of what we are doing. You
will see the drawing shows the three vector triangles corresponding to the three legs each
superimposed onto the same wind speed vector.

ã Copyright 1998 Doug Gray, Heathcote, Australia 2233. All rights reserved.
Page 2
G/S1, G/S2 and G/S3 correspond to the recorded ground speed vectors. If the Wind speed
and direction are constant for the legs then the velocity vector triangles corresponding to the
recorded ground speeds can be drawn on top of this single common Wind Speed vector W/S.
The tails of the G/S vectors each will lie on a circle with a radius corresponding to the True
Airspeed.
Since only three points are required to define the circle, three tracks are required to give a
solution for TAS.
Computation
Rather than providing a derivation I have reduced the solution to a series of spreadsheet
equations so that it may be of some practical value. These Excel equations can probably be
extended to any alternate spreadsheet software without too much difficulty.
The wind direction result is the direction from which the wind is coming just like the weather
report. The vector direction of the wind is at 180° from this.
A B C
1 Speed1 140 140
2 Track1 192 192
3 Speed2 112 112
4 Track2 283 283
5 Speed3 120 120
6 Track3 20 20
7 X1 =B1*SIN(PI()*(360-B2)/180)
8 Y1 =B1*COS(PI()*(360-B2)/180)
9 X2 =B3*SIN(PI()*(360-B4)/180)
10 Y2 =B3*COS(PI()*(360-B4)/180)
11 X3 =B5*SIN(PI()*(360-B6)/180)
12 Y3 =B5*COS(PI()*(360-B6)/180)
13 M1 =-1*(B9-B7)/(B10-B8)
14 B1 =(B8+B10)/2-B13*(B7+B9)/2
15 M2 =-1*(B11-B7)/(B12-B8)
16 B2 =(B8+B12)/2-B15*(B7+B11)/2
17 WX =(B14-B16)/(B15-B13)
18 WY =B13*B17+B14
19 Wind Speed =SQRT(B17^2+B18^2) 20.6
20 Wind Direction =MOD(540-(180/PI()*ATAN2(B18,B17)),360) 314.8
21 TAS =SQRT((B7-B17)^2+(B8-B18)^2) 130
22 Heading 1 =MOD(540-(180/PI()*ATAN2(B18-B8,B17-B7)),360) 200
23 Heading 2 =MOD(540-(180/PI()*ATAN2(B18-B10,B17-B9)),360) 287.8
24 Heading 3 =MOD(540-(180/PI()*ATAN2(B18-B12,B17-B11)),360) 11.7
Accuracy
Needless to say the aircraft should be flown on a constant heading with the speed stabilised,
and each recorded at the same power setting and same altitude. It would be good form to
ensure the aircraft was not in a slight yaw by making sure the slip ball is centred.
The actual tracks used are not critical, in fact any three tracks will give a TAS figure, but if
the headings are too close to each other then the TAS will be subject to error.
With a smaller difference in heading the greater will be the error in the final solution. This
can be appreciated by looking at fitting the circle to three points. The closer these three points
are then the more variability we will see in the size and position of the circle due to our
measurement errors.
Roughly speaking provided the three tracks differ from somewhere about 90° to 120° then
the resulting error in TAS will be of the same magnitude as the error in the GPS Ground
ã Copyright 1998 Doug Gray, Heathcote, Australia 2233. All rights reserved.
Page 3
Speed. An error in GPS speed of ±1 knot and ±1° in track will result in a corresponding error
of about ±1.3 knot.
Do not forget that other factors will also affect the accuracy. The most difficult to eliminate is
as a result of being in an ascending or descending airmass. This effect could be minimised
by testing in stable atmospheric conditions, or by collecting sufficient data points so as to
average out this error.
Determine your Compass Deviation as well.
In the spreadsheet equations I have included three lines titled 'Heading'. These correspond
to the actual headings for each of the legs or in other words the direction for each of the TAS
vectors.
If the compass heading is noted with the GPS data then over a period of aircraft
assessment plotting compass heading against the corresponding computed TAS Heading will
enable a deviation chart to be compiled for the aircraft. This of course assumes that the GPS is
set to display Magnetic track data.
While a conventional compass swing will provide such data very quickly and conveniently,
the opportunity to get such data for nothing is hard to resist.