Technology Notebook ContentsEffective Date: 15 June 98
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In the early 1970's, the AFFTC (Air Force Flight Test Center) developed a new method to calibrate airspeed. The method was originally dubbed the cloverleaf method due to the pattern prescribed in the sky. The idea is as follows. One assumes that wind remained constant while the aircraft performed consecutive turns to produce three passes through a common air mass. Ideally, the passes should be equally spaced heading wise (or 120 degrees apart) and at the same indicated airspeed. Besides the two components of wind (N & E) there would be an unknown error in true airspeed that would need to be computed. This paper will present the mathematics of this method and some substantiating data. They involve the solution of three nonlinear equations in three unknowns. It does not require that each pass to be executed at the exact same airspeed at precisely 120 degrees apart. The National Test Pilot School (NTPS), for instance, uses a method where the passes are 90 degrees apart, making the much simpler.
The development that makes this method dramatically more economical for flight test is GPS (Global Positioning System). The method has been applied with reasonable success by the NTPS. What this paper contributes beyond what NTPS has done is the nonlinear mathematical solution such that the test points do not have to be flown as precisely.
1Aircraft Performance Engineer, AFFTC, retired
2Aircraft Performance Engineer, AFFTC
3Flight Dynamics Engineer [Airspeed Calibration Specialist], AFFTC
4Flight Dynamics Engineer and Instructor, NTPS
NOMENCLATURE
# - Number
- aircraft heading from true North
- increment (correction to be added)
- ratio of specific heats = 1.40
- temperature ratio = Ta/TSL
(C - degrees Centigrade
(F - degrees Fahrenheit
(g - ground speed direction from true North
(K - degrees Kelvin
P/qcic - position error parameter ((P = Ps - Pa)
(w - wind speed direction from true North
a - speed of sound (knots)
AFFTC- Air Force Flight Test Center
aSL - standard sea level speed of sound = 661.48 knots
deg - degrees from true North
E - East
GPS - Global Positioning System
H - pressure altitude (feet)
Hg - Mercury
In - inches
knot - nautical mile per hour
M - Mach number
m - meter
Milli-inch - 0.001 inch
N - North
NTPS - National Test Pilot School
Pa - ambient pressure (psf)
Ps - static pressure (psf)
psf - pounds per square foot
PSL - standard sea level pressure = 2116.22 psf
Pt - total pressure (psf)
qcic- indicated dynamic pressure
S - South
Ta - ambient temperature ((K, (C or (F)
TSL- standard sea level temperature = 288.15 (K
Tt - total temperature ((K or (C)
Vc - calibrated airspeed (knots)
Vg - ground speed (knots)
Vt - true airspeed (knots)
Vw - wind speed (knots)
Vwx - x (or N) component of wind speed (knots)
Vwy - y (or E) component of wind speed (knots)
W - West
INTRODUCTION
This paper will not discuss the theory and operation the GPS system. In addition, it will not discuss air data systems at any length. Both subjects have been written about at length. The references and bibliography contain just a few of the numerous information sources on these topics. For the sake of this paper, the primary piece of information required of GPS is the accuracy of the velocities and at what update rate they are available. The military specification for velocity is 0.10-m/sec (0.19-knot). The data, in this paper, was available at one sample per second. The readability of the civilian GPS used for the NTPS tests was one knot.
This paper will attempt to explain and demonstrate the validity of a method to calibrate true airspeed (Vt). The method invokes the principle that the vector sum of ground speed plus wind speed is equal to airspeed. The terminology "true" airspeed is used to avoid the confusion with the cockpit indicator readings which are referred to as "calibrated" airspeed (Vc). For those not familiar with calibrated airspeed, the cockpit airspeed indicator only measures actual airspeed on a standard day (59(F) at sea level standard pressure (2116.22 psf). The cockpit indicator, historically, could be constructed mechanically with only one pressure input. That input was a differential pressure between total and static pressure. The true airspeed, Vt, on the other hand, was more complex. Vt requires computations involving total pressure (Pt), static pressure (Ps) and total temperature (Tt). The equations for Vt (for Mach (M) < 1.0) are included at the end of this paper.
By solving three equations in three unknowns, it will be shown how one can derive the unknown error in Vt and the North and East components of wind. Since it is easier to relate to wind speed magnitude (Vw) and direction ((w), the North and East components will be converted to magnitude and direction.
THE FLIGHT MANEUVER
The test is performed by first collecting stable data along a heading of (1. Only a few seconds of data are required to acquire average airspeed and ground speed data. Then a right-hand turn to a heading of (2 is accomplished and repeats another data collection. A final right-hand turn ends up at a heading of (3 and a final collection of data. The whole sequence should be performed in one continuous sequence. Left-hand turns could also be used. In that case, the heading sequence would be 1,3,2 instead of the 1,2,3 sequence for the right hand turns. The aircraft was flown on heading, but the data reduction involves track angle. Heading is the direction the aircraft is pointing while track is angle of the aircraft ground speed vector. Heading could also be considered the direction of the true airspeed vector when the sideslip angle is zero.
DATA PRESENTATION (AFFTC Test Procedure)
On August 19, 1997 three multi-track runs were performed using the AFFTC F-15 pacer (F-15B USAF S/N 132). A discussion of pacer aircraft can be found in references four and five. These runs were performed at nominal indicated conditions of 30,000 feet pressure altitude and indicated Mach numbers of 0.6, 0.7 and 0.8. Each run consisted of three separate passes at track angles of about 120( apart. In round numbers the first pass was at a track angle of 15( (N-E quadrant). Then a left-hand turn was performed bringing the aircraft around to a track angle of 255( (S-W quadrant). Finally, a second right-hand turn was performed to a track angle of 135( (S-E quadrant). Notice that the headings are separated by the ideal value of 120(. If the data are acquired at roughly equally spaced angles, then the method should produce reasonable results. The NTPS, in fact, has demonstrated that a separation of 90 degrees produces quite adequate results.
ERROR ANALYSIS
This method is a true airspeed calibration method. There are five measurements: total pressure (Pt), static pressure (Ps), total temperature (Tt), ground speed (Vg) and track angle ((g). The first two measurements come from pressure transducers. The third one is from a total temperature probe. The last two parameters are either GPS or Radar "measurements". The quotes are around measurements since the velocity and heading data are computed from derivatives of positions. The laboratory calibration accuracy for pressure transducers is about ( 0.001 in. Hg (0.071 psf) and about ( 0.10 (K for temperature probes. So, one will use these numbers and pick a typical condition near the test conditions of the data shown in this paper. Let:
Mach number = 0.800
Pressure Altitude = 30,000 feet
Ambient Temperature = 242.0 (K
At those conditions (and carrying out computations to beyond usual resolution):
Pt = 957.944 psf
Pa = 628.432 psf
Tt = 272.98 (K
Vt = 484.959 knots (true airspeed)
Since we are working with two different units on pressure, the conversion factor is as follows:
Inch Hg = 70.726 psf
Add 0.001 in. Hg "error" to Pt:
Pt = 958.0147
Computing true airspeed:
Vt = 484.985 knots
The error in computed true airspeed for an error in total pressure then is:
((Vt )/(( Pt) = (484.985-484.959)/(958.0147-957.944) =
= 0.580 (knots/psf)
Or = 0.042 knots per 0.001 in. Hg Total Pressure
Hence, for the laboratory accuracy of one-milli inch of Mercury (0.001 in. Hg) the error in total pressure results in a 0.042-knot error in true airspeed. Keep in mind this is the error slope at just this one set of conditions.
To examine ambient pressure errors, add the same error (0.001 in. Hg) to ambient pressure, while keeping the other parameters the same.
Pa = 628.5027
Vt = 484.902
Then,
((Vt /( Pa ) = (484.902-484.959)/(628.5027-628.432)
= -0.808 (knots/psf)
Or = -0.057 knots per 0.001 in. Hg Ambient Pressure
A 0.1-degree error in total temperature produces a true airspeed error as follows.
Vt = 485.048
((Vt /(Tt) = (485.048-484.959)/(0.1) = 0.89 (knots/deg K)
Or = 0.089 knots per 0.1 (K Total Temperature
For this particular flight condition, an error in the aircraft parameters equal to their laboratory accuracies would produce errors in Vt of less than 0.1 knot. For the AFFTC data, some of the results will be presented to greater than 0.1-knot resolution, but this does not imply that accuracy level has been achieved.
Errors in ground speed will produce errors in true airspeed proportional to the error in the ground speed on each leg of the method. The ground speed error is likely to be just the readability of the data. In the case of using a hand held GPS unit, the error in each leg might be either to the nearest knot or to the nearest one-tenth of a knot. The military specification accuracy of GPS velocity is 0.1 meter per second (0.19 knot).
AFFTC TEST DATA Set Number One
The results for the August 19, 1997, data are summarized in the following tables.
THE INPUTS:
Aircraft average measurements and parameters computed from the measurements:
Note: the subscript i denotes indicated value
INERTIAL SPEEDS (GPS):
Notes: Subscripts a, b, and c denote separate passes
Runs 2a & 2b used Radar Data
THE OUTPUTS:
The pacer corrections are known to a high degree of accuracy. These corrections are in the form of a curve of the parameter (P/qcic versus indicated Mach number. These are corrections that are applied to pacer data any time the pacer is used to calibrate another aircraft. The following is the average and standard deviation difference between these three multiple track data points and the pacer curve.
Ground speed time histories for run number one are depicted in the plots below. Run number one is actually three separate passes (1a, 1b & 1c) at the same aim airspeed but at different ground speeds. These compare radar data and GPS data, both of which have been smoothed in this case with a 19-point second order polynomial curve fit.
The above plots are time histories of ground speed for the first three of nine passes. The next plot compares un-smoothed GPS ground speed to the smoothed GPS ground speed. As of this writing, the level of "noise" in the un-smoothed data remains unexplained.
SUMMARY OF TIME HISTORY DATA (Average Values):
Note: (GPS = GPS-Radar
For the first run (number 1a), the following plot illustrates a comparison of true airspeed. The pacer aircraft has a direct output of corrected true airspeed. That is shown as series number 2 in the plot. Series number 1 is a computation of true airspeed from GPS ground speed plus the computed wind speed.
As a check on the data, a comparison was made with weather balloon data. The balloon data were from about two hours after the completion of the last test point. The following table shows a comparison of the balloon data with the aircraft test data.
The temperature and wind heading numbers agree easily within the accuracy level of the balloon data. However, there was a severe wind gradient (change of wind speed versus altitude) during that test day at the test altitude of 30,000 feet. The following plot of the wind speed is a graphic depiction of wind gradient.
By choosing to illustrate only the winds surrounding the test altitude, one has missed some even more interesting data. There also seemed to be a wind speed inversion. By adding just the points above and below the ones shown above, a different picture emerges.
From the start of the first pass (1a) to the completion of the last pass (3c) was 37 minutes. It seems clear that something considerably less than a full minute of data on each pass would be quite adequate. A ten-second average would suffice. Then, by relaxing the requirement to maintain the test airspeed exactly, an additional amount of test time could be saved. Without the need for radar tracking it becomes unnecessary to co-ordinate with the radar tracking team and that saves even more time. It seems reasonable that a factor of two or more savings in flight time could be achieved. Thus, not counting the time required to climb to the test altitude, each set of three passes could be concluded in about five minutes or less.
An interesting observation is that as long the error in airspeed is the same on each leg, the computed value of wind will be identical. That means one could use this technique to "measure" winds aloft even without having a precise airspeed calibration. Quotes are around measure since one would compute the winds rather than measure them directly.
AFFTC DATA Set Number Two
Additional data was obtained on April 3, 1997 using the AFFTC F-16B (S/N 633) pacer. There was no GPS data available. However, radar tracking was used. The atmospheric conditions (as shown below) were less than ideal and still the data were acceptable. The onboard computer calculates corrected values for Vc, H, and Mach, so rather than comparing to the position error curves, a direct comparison to these corrected values is shown in the following table.
The data were recorded at a nominal altitude of 30,000 feet pressure altitude. The delta's are corrections to be added.
The parameter one "measures" with this method is the error in true airspeed ((Vt). The other parameters are computed assuming zero total pressure error. It is, however, not essential to assume there is no error in the measured total pressure. All one knows for sure is that there is a given error in computed true airspeed. That true airspeed is calculated from all three of the aircraft air-data parameters (total pressure, static pressure and total temperature). For the purposes of this paper the assumption was made that all the error in true airspeed could be attributed to an error in static pressure. One needs to recognize the potential for errors in the other two measurements.
Notice the magnitude of the winds - in excess of 100 knots. The average winds at Edwards at 30K are about 50 knots. The winds on the cloverleaves flown by the F-15 were about 45 knots. There was no weather balloon data available for the date of this flight (3 April 1997). The data from the day prior showed winds at 30,000 feet at about 20 knots at 25 degrees heading, and data from the day after indicated winds of 90 knots at 353 degrees.
NTPS DATA
On March 6, 1998, the NTPS flew a set of five multi-track maneuvers on a medium transport aircraft (Merlin III) shown above. Previous tests on the same aircraft were reported in reference 3. In this case, the data were all hand recorded from cockpit instruments and a hand-held GPS unit. The data were read to the nearest knot. The flight pattern was not the cloverleaf pattern used by the AFFTC but a right angle pattern. The first pass was at some heading (1. This was followed by a right hand turn to intercept the path of the first pass at a right angle to a heading of (2. That makes (2 = (1 + 270 degrees. Finally, a third pass is conducted after a right hand turn through an additional 270 degrees. That would make the heading of the third pass ((3) exactly 180 degrees from the first pass at heading of (1. The data were all at an indicated pressure altitude of 8,000 feet. The data were single values, hand recorded, as opposed to the AFFTC data that were averaged at one sample per second over a full minute. From start to end, the total testing time was 40 minutes. The following data were recorded. Instrument corrections were applied to the indicated airspeed. Instrument corrected altitude became 7,990 feet.
Processing the data through the same software used for the AFFTC data yields the following results.
The error in calibrated airspeed is presented in the following plot.
Presented in the plot are data from the March 6, 1998 flight as well as data from a flight on March 18, 1997 (reference 3). The cone data were collected concurrently. Cone refers to data using a "trailing cone" and a "Kiel tube". The trailing cone is a pressure probe protected by a shield that is trailed behind the aircraft to measure ambient pressure. The Kiel tube is a total pressure probe that is has very small errors due to angle of attack. These two devices are discussed in more detail in reference four. The significance of the above data is open for analysis, but the errors are within the readability of the gauge. A null correction would probably be adequate.
The mathematics used to solve for the unknown errors in true airspeed and for the wind components for all the data in this paper is described in the following section.
Mathematics of the Multi-Track Method
The basic vector equation that one will solve for the multi-track method is nothing more than true airspeed equals the vector sum of ground speed and wind speed.
where the N & E components of ground speed are either direct outputs of the GPS or are computed as follows:
The aircraft track angle (or the direction of the ground speed vector) is (g. Writing down the relationship that true airspeed squared is equal to the sum of the squares of its components.
Substituting (2), (3) and (4) into (5) yields:
Multiplying out (6) and collecting terms, one gets:
Defining the following:
Each pass produces an equation. Subscript 1 is the first pass, 2 is the second and 3 for the third. The unknowns x, y & z are presumed constant for all three runs. In matrix form, the equations are as follows:
In matrix shorthand form:
The vector of unknowns X is solved by multiplying each side of (9) by the inverse of the [A] matrix.
The unknowns x, y and z in the {X} are also contained in [A]. So an iteration is required. The initial estimates for the X values will be zero. Then equation (10) is used to compute a new set of X values. These are inserted into [A] and is inverted again and equation (10) is used again. The process is repeated until convergence occurs.
When the iteration is complete you have solved for the desired numbers, namely an error in true airspeed and two components of wind.
COMPUTING INDICATED TRUE AIRSPEED
Three measured parameters are required: total pressure (Pt), ambient pressure (Pa) and total temperature (Tt). Using English units, the corresponding sea level standard values of temperature and pressure are:
PSL = 2116.22 pounds/foot2
TSL = 288.15 ( Kelvin
aSL = 661.48 knots (nautical miles per hour)
= 1116.45 feet per second
Where a is the speed of sound
Where:
= 1.40 (ratio of specific heats)
R = 3,089.8 ft2 per (sec2 (Kelvin)
[Universal gas constant for air]
= Temperature ratio
= Ta /TSL
The method has only been applied at subsonic speeds. However, there is no fundamental reason that the basic method would not work as well supersonically. The problem is the huge fuel expenditure in calibrating an aircraft at supersonic speeds. To compute Mach number (M) at subsonic speed use the Bernoulli equation:
Substituting the 1.40 value for :
Or, solving for Mach number:
Now that one has Mach number, one can compute the ambient temperature and then the indicated true airspeed.
Finally, one can calculate the indicated true airspeed. The indicated true airspeed is a true airspeed computed before applying any "position error" [or corrections to total pressure, static pressure or total temperature].
From the iterative solution of the three non-linear equations in three unknowns (equations 6,7, & 8) one can compute the unknowns:
The true airspeed from this method is simply as follows:
With an improved value of true airspeed, one can compute a better value of true Mach number.
One is still not quite finished, since the calculated speed of sound is a function of the true Mach. It is necessary to iterate between equation 16 and equation 21 for the best value of the true airspeed & Mach number.
At this point, one has a choice of taking equation (13) and solving for either total pressure or ambient pressure. Most flight test aircraft have a total pressure probe, and zero total pressure error is assumed. In other cases, the multiple track method is used to calibrate a pacer aircraft (with a total pressure probe). The pacer aircraft can then be used to calibrate aircraft where both the total and static pressure and the total temperature are all unknowns.
More than three passes
As the title implies, one may conduct multiple passes. The suggestion is to simply divide 360 degrees by the number of passes to determine the angular separation between the different passes. For instance, four passes would be 90 degrees apart. Five passes would be 72 degrees apart, etc. Then, various sets of three data points would be individually processed and the results would be averaged to produce a "best estimate" of the airspeed error at the average test condition. By having more than the minimum three points, one can assess the statistical confidence in the data point using something as simple as the standard deviation.
SUMMARY
The AFFTC developed this method more than a quarter century ago, and it has yet to be publicized. The equations and software were not documented until 1976 (reference 1) in an internal memo. The concept was not very dramatic. It was simply that the airspeed is equal to the ground speed plus the wind speed. Where it got complicated was when the aircraft velocity vector (or it's track over the ground) was in a different direction than the wind. Unfortunately, this is always the case. Even back in 1970, we could measure with a high degree of accuracy the aircraft velocity and direction using radar. That meant dollar expense and delay in data turnaround. In addition, it required ground control coordination that increased the flight time. Therefore, the method was costly. It was only used as a calibration check for the AFFTC pacer aircraft. No attempt was made to sell the cloverleaf method to any of the other test forces as a supplementary airspeed calibration method. The method, which was then called the cloverleaf method, would not supplant the other traditional methods. It would provide additional information to validate the airspeed calibration on the test aircraft. With the advent of GPS, the method becomes very economical. So economical, so economical, in fact, that it would become the primary method of calibrating true airspeed. Keep in mind, however, that it determines true airspeed and then computes an error in true airspeed. From true airspeed error, one must determine what is the source of the error. The usual assumption is that all of the error is in static pressure. Therefore, what this method does is add an additional method to improve the quality of the overall calibration of the air data system.
REFERENCES
Reference 1: Wayne M. Olson, "True Airspeed Calibration Using Three Radar Passes", Performance and Flying Qualities Branch Office Memo, Air Force Flight Test Center, CA, August 1976.
Reference 2: J.A.Lawford and K.R.Nipress, "Calibration of Air Data systems and Flow Direction Sensors", pages 16-20, AGARD AG-300-Vol.1, September 1983.
Reference 3: Gregory V. Lewis, "A Flight Test Technique Using GPS For Position Error Correction Testing", National Test Pilot School, Mojave, CA, July 1997.
Reference 4: William Gracey, "Measurement of Aircraft Speed AND Altitude", John Wiley & Sons, 1981.
Reference 5: Albert G. DeAnda, AFFTC Standard Airspeed Calibration Procedures", AFFTC-TIH-81-5, June 1981. Bibliography
David Fox, "Is Your Speed True", KITPLANES Magazine, February 1995.
Richard Dwenger, John Wheeler, James Lackey, "Use of GPS For An Altitude Reference Source For Air Data Testing", presented at 1997 Society of Flight Test Engineers Symposium.
Ralph Kimberlin and Joseph Sims, "Airspeed Calibration Using GPS", AIAA 92-4090, presented at 6th Biennial Flight Test Conference, August 24-26, 1992.
HYPERLINK http://www.navcen.uscg.mil/gps http://www.navcen.uscg.mil/gps
Bill Clarke, "Aviator's Guide To GPS", TAB Books, 1994.
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