Jan 10, 2005 - minutes and covering about 10 nautical miles. An argument used today against this method is that it is too time consuming. It is not if...

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An Informal Memo by Wayne Olson .....................................................................1 Déjà vu – I’ve seen this before:.............................................................................1 Introduction ...........................................................................................................2 Ground Speed Course – Errors Due to Crosswind ...............................................3 The Three-Vector Method Is Simply a Better Way ............................................5 History of the Concept ..........................................................................................5 “Cloverleaf” Computations with 1927 Airship Data................................................8 Airspeed Measurements by Germany In 1918....................................................10 True Airspeed Calibration Using Three Radar Passes .......................................11 AFFTC F-15 Data ...............................................................................................12 AFFTC F-16B Pacer Data...................................................................................13 Cessna 180 - Airspeed Calibration Using GPS...................................................14 Embraer EMB 140 Cloverleaf Data.....................................................................16 The Doug Gray Algorithm ...................................................................................22 Gray Algorithm Comparison Using Cessna 180 Data .....................................22 Gray Algorithm versus F16 Data .....................................................................23 Finally .................................................................................................................23 References .........................................................................................................24 List of Figures and Tables...................................................................................25 Abbreviations ......................................................................................................25 Symbols ..............................................................................................................26

Déjà vu – I’ve seen this before: Since this is an informal memo, I feel free to give a short personal note about a discovery I made recently. While in the process of downloading a number of old NACA documents which are available for free download at www.larc.nasa.gov , I found something that I had seen before (as in déjà vu). Let me take you back to early in my career at Edwards. It was early 1970s. My 1

supervisor at the time was Willie Allen. The airspeed calibration guy for the center was Al DeAnda. [Al retired in 2002 after 50 years in federal service and 46 years of that at Edwards]. Anyhow, I noticed Willie and Al in Willie’s office with this huge sheet of paper and various large drawing tools. Huge as in 4 to 6 feet square. On that was drawn a circle. Inside that circle were three vectors. The vectors were drawn to scale with appropriated angles. The lengths corresponded to ground speed and the angles were course over the ground. There were three of these vectors. This was actual data from three radar tracked runs with the AFFTC pacer aircraft. From this, they figured out true airspeed and wind speed and direction. It just looked awfully hard – there just had to be a better way. After all we had computers now. Some time later in a math class at the Test Pilot School, we were learning about vectors and matrices. It struck me - 3 equations in 3 unknowns! It was then just a matter of formulating those 3 equations. So, they were nonlinear. Nonlinear in the sense that the unknowns (2 components of wind and an unknown error in true airspeed) were on both sides of the equations. Iterate! I learned that in college. The déjà vu part came when I came across a circle with speed vectors that looked identical to Willie and Al’s circle drawn almost 50 years later. In History of the Concept, page 5, you will see the NACA drawing.

Introduction This memo will recommend that the FAA accept a GPS Three-Vector Method as a viable airspeed calibration method for aircraft certification. In a 1998 Society of Flight Test Engineers symposium paper1, the mathematics for one such three-vector method is detailed. In that paper, the references list a few other variations on the method. To our knowledge, all of the other methods make assumptions about the angles in order to solve the equations. Doug Gray2 makes no such angle assumption, however he does assume that all three true airspeed vectors fall within a perfect circle. No such assumptions are made here; the equations are solved in a nonlinear iteration. The details are contained in the paper – here I will present a summary. The paper also contains three sets of data collected to illustrate the method. We initially developed the technique to calibrate the Air Force Flight Test Center pacer aircraft, where we used Radar to measure ground speed and direction. Al DeAnda and Willie Allen of AFFTC developed the flight test technique and I created the computer software. The technique was first described in an unpublished Edwards AFB office memo8, which I will discuss later in this document. Our method was also described in a 1983 AGARD document3; however, the AGARD document did not include our mathematical algorithm.

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The method is a true airspeed method. GPS, even without using differential GPS, provides very accurate velocity information in the horizontal plane. The accuracy is on the order of ± 0.1 knot or better. [I should add that the resultant true airspeed accuracy is not necessarily accurate to the same ±0.1 knot.] It is not necessary to acquire long time histories of data – a few seconds average will be sufficient. The GPS velocity, however, is inertial or a ground speed. What we would like is true airspeed. The basic relationship in the horizontal plane is that true airspeed ( Vt ) is ground speed ( Vg ) plus wind ( Vw ). These three speeds are all vectors with both magnitude and direction. We will work with these in the north and east direction. GPS provides (or we can compute) north and east components of ground speed. The aircraft air data system gives us an indicated true airspeed (we may have to compute that from other parameters). The indicated true airspeed ( Vti ), will have some unknown error ( ∆Vt ). The wind speed components in the north direction ( VwN ) and in the east direction ( VwE ) are unknowns. The equation we will be solving is (1.1).

(V

ti

+ ∆Vt

) = (V 2

gN

+ VwN ) + (VgE + VwE ) 2

2

(1.1)

By flying in three distinctively different directions (ideally about 120 degrees), we can expand (1.1) to three equations in three unknowns. These are three nonlinear equations in three unknowns. They are nonlinear in the sense that no matter how hard I tried; I could not come up with equations where the three unknowns appear only on one side of the equations. An iterative solution is easy to do now with a PC. I recently discovered that David Gray came up with a solution (See: The Doug Gray Algorithm, page 22). Others have been able to derive exact formulas for special cases. One of these is where two passes are exactly 180 degrees apart and the third is perpendicular to the first two. This has been called the “horseshoe heading” method. These also require that the airspeeds are identical for each pass. These are both viable and quite acceptable techniques. One just needs to be aware of these limitations. 4,5

Ground Speed Course – Errors Due to Crosswind I would suggest using this Three-Vector method to replace the venerable ground speed course. Unless you have an INS and are able to estimate the magnitude of the crosswind and correct for it, there will always be some error in the ground speed course method. An example will illustrate the effect. The example will use what some would say is an extreme case. However, this wind example is very near a common wind speed at the AFFTC. Let us assume we 3

are flying a track of due North at 100 knots indicated true airspeed. We shall assume we have computed the true airspeed from our Pitot-static parameters. Then, presume a wind speed of 20 knots at 90 degrees. Wind direction is “from which the wind is blowing”, so the wind is blowing from due east. We now have a speed triangle as shown in Figure 1. Then, we turn and go down our ground speed due South. For simplicity, we are assuming our ground speed course is laid out due North-South. Figure 2 shows the reverse run.

Figure 1

Run 1 of Ground Speed Course

Figure 2

Run 2 of Ground Speed Course

Then, from our triangles, we can compute the two groundspeeds. Vt12 = Vg12 + Vw12 → Vg1 = Vt12 − Vw12 = 1002 − 202 = 98.0 knots

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(1.2)

Vt 2 2 = Vg 2 2 + Vw 2 2 → Vg 2 = Vt 2 2 − Vw 2 2 = 1002 − 202 = 98.0 knots

(1.3)

By the ground speed course method, the true airspeed is the average of the ground speeds or 98.0 knots. In this case, we incurred an error of 2.0 knots due to the method. That is with a measurement of ground speed using GPS where the GPS accuracy is of order of 0.1 knots. We do not incur any error due to the method with the Three-Vector method.

The Three-Vector Method Is Simply a Better Way The crosswind problem completely disappears. In fact, you end up with both speed and direction for the wind as well as the error in true airspeed. The accuracy of GPS (even a handheld one) is at least ± 0.1 knot. This is an accuracy I have demonstrated on several projects that are discussed in this memo.

History of the Concept In 1927, the NACA used a method they referred to as a triangle method to calibrate airspeed systems on an airship6. Figure 3 is a speed diagram of their data extracted from the NACA report.

Figure 3

Graphical Solution for Airspeed and Wind speed.

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The technique described by NACA is precisely the technique for the GPS Three-Vector Method. In lieu of GPS, they timed the vehicle over a known distance to get groundspeed. The known distances also had a known compass heading. From compass heading and magnetic declination, one can compute the true ground track. The local declination in 1927 was (-11) degrees. GPS provides the same information, namely groundspeed and true ground track. Figure 4 is a portion of the NACA description.

Figure 4

NACA Test Technique Description

The key part to note here is that they determined “the wind was nearly constant at 10.2 knots from the southwest, and ship’s airspeed was 58.4 knots”. Next is the data they collected in Figure 5.

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Figure 5

NACA Speed Data

The original software programmed in FORTRAN was created to reduce Three-Vector airspeed calibration data. In early 1970s, method was used to calibrate the AFFTC pacer aircraft using Radar. In the late 1990s, the same algorithm was used to calibrate AFFTC pacers, however this time the ground speed and track angles were provided by GPS. In addition, the new software has been reprogrammed in Excel. Since the pacers fly what would appear in

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the sky as a “cloverleaf” maneuver, we dubbed this the cloverleaf maneuver and the software is denoted as Cloverleaf. The airship did not fly a cloverleaf and lower speed aircraft would not – more like the triangle maneuver as described by NACA.

“Cloverleaf” Computations with 1927 Airship Data I ran the airship data through the cloverleaf Excel program. I did run one and run two as two separate runs. Table 1 contains the test data. The point of this exercise is to show that had they had GPS in 1927, they already had the maneuver and just needed the software that solves a set of nonlinear equations. I will use the “electric meter” as airspeed since it is the only airspeed of the four that is a true airspeed. The electric meter is a windmill device and labeled it propeller in Table 1. NACA, of course, preferred their Pitot-static instrument. The others are all Pitot-static instruments. Table 1

NACA Data Processed With Cloverleaf Program

Run No.

AC CB BA

Course (True) Deg 332 89 211.5

Leg

Length Feet 59,500 60,800 63,200

Time Sec 570 552 769

Speed Ft/Sec 104.39 110.14 82.18

Speed Knots 61.85 65.26 48.69

1 1 1 2 2 2

AB BC CA

31.5 269 152

63,200 60,800 59,500

548 718 648

115.33 84.68 91.82

68.33 50.17 54.40

NACA Propeller Airspeed Airspeed Knots Knots 59.5 59.0 60.0 59.0 59.5 58.8 59.5 60.0 59.5

58.0 60.0 59.5

I did the calculations of the speeds in feet/sec and knots from the length and time. They agree to within 0.1 ft/sec and 0.1 knot for the most part; however, a couple of points are farther apart – worst case of 0.4 knots. The rather small differences were more than what would be explained by the old nautical mile of 6,080 feet versus today of 1,852 meters (exactly), which comes to 6076.115 feet, for the units conversion of 0.3048 meters per foot. In addition, I wanted to avoid changing units in the software. From Table 1, one interesting note is that each leg was taking about 10 minutes and covering about 10 nautical miles. An argument used today against this method is that it is too time consuming. It is not if it is planned and flown well – you do not need a lot of data points to get a good average speed and heading. You will see how little time they can take in the Cessna 180 data later in this memo.

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The NACA report did not specify the altitude or the ambient temperature for these speed runs. They were flying above “a triangular course between Sandy Hook, Staten Island, and Rockaway”. With that, and map software by Microsoft, I have drawn an approximate course in Figure 6.

Figure 6

Course for Airship Trials

As can be seen it is over ocean, so we can at least hope for some reasonably stable conditions. To make things easy, I will assume standard day sea level conditions. Since we will be entering true airspeed, the atmospheric conditions should not influence the wind computations. However, the inputs to the software are total pressure, static pressure and total temperature. Each run, with three legs each, is one data point. The SFTE paper will go through the math in detail, but what we are doing is solving three equations in three unknowns. Each leg provides constants for one of the three equations. The three unknowns are two components of wind (north and east) and an error in true airspeed. Then, from the assumption that all of the Pitot-static system error is contained in static pressure, we can compute true values of Mach number, calibrated airspeed, and ambient temperature. We already know total pressure and total temperature since we assumed they had zero error. 9

So, what is the best estimate for true airspeed and wind for these six passes? They assumed a constant true airspeed for all six runs (the same constant) – that produces a perfect circle. From their graphical method (Figure 3), they came up with the following.

Vt = 58.4 knots Vw = 10.2 knots

ψ w = ("from the southwest") = 225° I obtained almost exactly what they got. Our results: 58.6 knots, 10.0 knots at 225 degrees – rounded to the number of digits they expressed. I computed this as two sets of data - run 1 and run 2. Then, I averaged those two results.

Airspeed Measurements by Germany In 1918 An even earlier determination of true speed using the same method was conducted by Germany7. They used two phototheodolities to track aircraft traveling at altitudes of 9,000 to 14,000 feet and speeds in the 80 to 90 knot range. Our Excel software reproduced their results almost exactly. They solved the speed triangles using the same graphical methods used by NACA. Figure 7 was extracted from the NACA translation of the German tests.

Figure 7

Velocity Diagram from 1918 (Germany)

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I used their symbology for the speeds and winds and converted to knots. Table 2 compares their results to ours. The first number is ours and the second is theirs. Table 2

Fig.13 Ve= Vw= Fig.14 Ve= Vw= Fig.15 Ve= Vw=

Airspeed Triangle Data from Germany in 1918

Vt= 45.0 Vw= 10.7

86.8 87.5 21.4 20.8

Fig.16 Ve=

Vt= 44.5 Vw= 16.5

86.2 86.5 32.3 32.1

Fig.17 Ve=

Vt= 43.2 Vw= 20.7

84.2 84.0 40.9 40.2

Fig.18 Ve=

Vw=

Vw=

Vw=

Vt= 44.7 Vw= 6.8

87.0 86.9 13.5 13.2

Vt= 43.4 Vw= 4.2

84.1 84.4 8.6 8.2

Vt= 43.3 Vw= 11.1

83.4 84.2 21.7 21.6

For instance, for run Fig.13, I computed an airspeed of 86.8 knots, while they came up with 45.0 m/sec, which converts to 87.5 knots. I got a wind speed of 21.4 knots and their wind speed converts to 20.8 knots. That was excellent agreement – given some room for error on our part, as I needed to measure the ground track angle graphically from their diagrams (as in Figure 7). For the most part, our airspeeds were within one knot at just over 80 knots.

True Airspeed Calibration Using Three Radar Passes In the early 1970s at the AFFTC, we essentially reinvented the wheel developed first by the Germans in 1918. At the time, this author was not aware of the German tests. We used Radar to calibrate the AFFTC pacers8. To do the computation, instead of a rather laborious graphical solution, an iterative computer algorithm was developed. That algorithm along with diagrams and actual test data are included in the 1976 office memo. Nearly the identical algorithm was converted to an Excel version to process GPS data1 some 20 years later. The original memo presented nine data points all collected on the same flight at same altitude and the same airspeed. The goal was to show repeatability

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and accuracy of the method. Table 3 presents the data in a different form than in the early memo. The nine runs will be grouped in blocks of three. Table 3

Summary of Radar track data

Run

Vti

Vg

σg

∆Vt

Vw

ψw

No. 1 2 3

Knots 462.2 461.9 463.5

Knots 473.0 466.0 451.0

Deg 149.3 256.4 21.9

Knots 0.15 0.15 0.15

Knots 13.2 13.2 13.2

Deg 4.8 4.8 4.8

4 5 6

462.1 463.3 463.3

469.8 468.4 454.8

131.5 253.5 15.8

1.55 1.55 1.55

10.3 10.3 10.3

4.4 4.4 4.4

7 8 9

463.1 464.8 464.8

470.0 466.0 455.8

137.6 252.7 19.9

-0.57 -0.57 -0.57

9.3 9.3 9.3

354.3 354.3 354.3

Each set of three runs will produce a set of the three unknowns ( ∆Vt , VwN and VwE ). The wind speed magnitude ( Vw ) and direction (ψ w ) are then computed from the components. There was a substantial difference in the speed error term between these three sets. Without access to the original data, it is impossible to determine the source of the problem. The positive way to look at the results is that the errors were well less than 1% of the true airspeed.

AFFTC F-15 Data One of the three data sets I presented in the SFTE paper1 was on the AFFTC F-15B pacer aircraft. The data were flown in August 1997 at nominal conditions of 0.6, 0.7 and 0.8 Mach number at 30,000 feet pressure altitude. Because we wanted to compare GPS to Radar, test coordination time added significantly to the total test time. These three cloverleaf points consumed about 40 minutes of flight time. In addition, each pass was a full minute long. Without these two factors, we would be able to collect this data in much less time. These tests were on the previously calibrated F-15B pacer. We had a high degree of confidence in the calibration curves. One lesson we learned on this data set is that we need an accurate total temperature. Initially, the data showed rather large errors versus the pacer curves – on order of 3 knots. After the temperature probe was recalibrated, the errors were dramatically reduced. We did not lose the data as we did the temperature calibration post flight.

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Table 4

Summary of F-15B Pacer Cloverleaf Data

Run

Vg

σg

Vti

ψw

Vw

∆Vt

∆Vt PACER

∆Vt Ref 1

No. 1a 1b 1c

Knots Deg Knots Knots Deg Knots 409.71 18.40 360.76 47.97 225.21 6.78 326.49 257.73 361.07 47.97 225.21 6.78 370.24 127.17 359.85 47.97 225.21 6.78

Knots 6.75 6.75 6.75

Knots 6.07 6.07 6.07

2a 2b 2c

471.24 16.48 420.13 390.55 258.03 420.68 430.80 127.75 421.55

8.76 8.76 8.76

8.38 8.38 8.38

8.94 8.94 8.94

3a 545.06 16.75 493.39 46.17 223.39 10.83 3b 465.86 257.23 494.03 46.17 223.39 10.83 3c 506.76 128.24 493.89 46.17 223.39 10.83 Notes: • The output parameter for airspeed error is ∆Vt

10.55 10.55 10.55

10.87 10.87 10.87

• •

47.25 47.25 47.25

221.50 221.50 221.50

The next column ( ∆Vt PACER ) is the airspeed error from the pacer curves Last column is the airspeed error presented in Reference 1.

That last note deserves some explanation. In processing this 1997 data, I could not precisely reproduce the results in the SFTE paper1. I did do some additional processing of the data after the submittal deadline of the paper. Anyhow, the results in the paper were slightly different (larger errors) – however the worst case difference was less than one knot in true airspeed. One might wonder how we were able to achieve airspeed accuracies of better than one knot at speeds between 360 knots and 500 knots. First, we had precision pressure transducers that were calibrated to better than ± 0.001 in Hg. At 300 knots calibrated airspeed at 30,000 feet, an error of 0.001 in Hg in static pressure comes to 0.02 knots. Then GPS, even without differential GPS is a ± 0.1 knot error system. However, that is on any given data point. By averaging over even as little as 10 seconds, one should have a fraction of 0.1 knot accuracy.

AFFTC F-16B Pacer Data A second set of data in the SFTE report1 was with the AFFTC F-16B pacer. On this data at the same conditions as the F-15, however, we had very high winds (over 100 knots) and had to use Radar instead of GPS. Despite those problems, we still had results where the true airspeed errors were less than 1% of the pacer data. I felt that were the GPS working and we did the maneuvers quicker, we would have gotten better results. The data, all at 30,000 feet, were flown in April 1997. The true airspeed from the pacer already has corrections

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applied. Then, we are essentially seeing how close we can get with this cloverleaf data. The parameter we would like to be near zero is the true airspeed correction to be added ( ∆Vt ). The results are shown in Table 5. Table 5

F-16B Pacer Cloverleaf Data Summary

Run

Vg

σg

Vt

∆Vt

Vw

ψw

No. 1a 1b 1c

Knots 415.3 421.4 247.4

Deg 241.3 146.5 16.4

Knots 356.3 356.5 358.4

Knots -2.5 -2.5 -2.5

Knots 108.7 108.7 108.7

Deg 12.1 12.1 12.1

2a 2b 2c

306.9 475.4 484.3

19.7 242.9 144.3

414.7 419.0 417.1

-0.5 -0.5 -0.5

108.3 108.3 108.3

10.4 10.4 10.4

3a 3b 3c

381.1 542.7 555.3

16.2 244.9 148.0

489.1 492.3 488.7

-4.1 -4.1 -4.1

104.3 104.3 104.3

11.2 11.2 11.2

Cessna 180 - Airspeed Calibration Using GPS On June 20, 2001, we had a flight in a Cessna 180 to collect airspeed calibration data. We flew out of Grove Field in Camas, WA. The test area was generally over Kelso, WA. This was not a formal test flight – just a flight that pilot Bob Elliot had graciously agreed to fly Paul Cannon and myself. My personal goal was to check out using a handheld GPS in flight test. By 2001, using GPS in flight test was old hat. However, this author had not done it as of yet. I had, however collected considerable data in a career at Edwards, AFB. We had planned to obtain cloverleaf data at three airspeeds (90, 100 and 110 MPH) and at three altitudes (3,000, 5,000, 7,000 and 9,000 feet). Since the aircraft climb speed is 90 MPH, the plan was to collect airspeed calibration data at the climb speed and 10 MPH on either side. However, due to a failure with my laptop computer we only obtained four out of the 12 planned points. Figure 8 shows the latitude versus longitude trace for the first run. This was a continuous maneuver – only the data points used in the cloverleaf program are shown. From the first data symbol on the plot to the last was under four minutes and the total time from first data point to last was less than one hour.

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Of significance here is these three passes were not flown in a cloverleaf pattern, but rather in a turn. Our pilot would stabilize on speed for about 15 seconds, then turn 120 degrees and stabilize again at same speed, then repeat a third time. Run 1.1 100 MPH at 9,000 feet 46.080 46.075

Latitude (deg North)

46.070 46.065

1.1 a 1.1 b

46.060

1.1 c

46.055 46.050 46.045 46.040 46.035 122.980 122.985 122.990 122.995 123.000 123.005 123.010 123.015 123.020 123.025 123.030 Longitude (deg West)

Figure 8

Latitude versus Longitude – First Run

Even though I did not compare the data collected from this cloverleaf method to another method, I did show that the Pitot-static errors were quite small. I did not have airspeed, altitude or total temperature recorded. We relied on pilot Bob to fly on condition. From a combination of his piloting skill and the fact on his old Cessna 180, he had some modern Avionics including an autopilot we obtained rather smooth data. The tests were all flown on the MPH needle of the airspeed indicator. However, the software is all in Knots, so beware of some mixed units. Then, keeping this memo short the summary of the numbers are in Table 6 and Table 8. The airspeeds and altitudes are pilot aim conditions.

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Table 6

GPS Inputs for Cessna 180

Run

Vg (1)

σ g (1)

Vg ( 2 )

σ g ( 2)

Vg ( 3)

σ g ( 3)

# 1.1 1.2 1.3 1.4 2.1

Knots 89.87 104.36 92.81 91.98 96.00

Deg 206.83 331.06 146.78 145.23 303.38

Knots 100.55 106.22 90.77 85.76 79.17

Deg 76.01 75.89 24.92 26.63 176.41

Knots 109.34 80.24 95.44 90.85 95.39

Deg 326.24 204.00 265.83 261.24 53.10

Table 7

Run

Cloverleaf Outputs for Cessna 180

H iC

ViC

∆VC

Vti

# Feet MPH MPH MPH 1.1 9,000 100.0 -0.9 115.6 1.2 7,000 100.0 -0.7 111.6 1.3 5,000 100.0 -1.2 108.3 1.4 3,000 100.0 -2.0 105.1 2.1 9,000 90.0 -0.2 103.8 Note: ∆V parameters are correction to be added

∆Vt

Vw

ψw

MPH -1.0 -0.8 -1.3 -2.1 -0.3

Knots 11.3 16.1 2.7 3.8 10.8

Degrees 174.3 207.3 60.0 13.4 176.4

This represents only a few runs; however, there are some confidence builders in the correctness of the data. The first would be that given that the readability of the airspeed indicator is at best to the nearest 1 MPH, the variation in the correction to be added is very small and a correction to be added of -1 knot would be quite reasonable. Then, notice that runs 1.1 and 2.1, which are at the same altitude produce computed wind speed, and direction that is only 0.5 knot apart and essentially identical wind direction. I should note also that from the beginning of the first run at 9,000 feet to the end of the last run, also at 9,000 feet the total elapsed time was 55 minutes. This was the total clock time, not just the test time. Each run identified in the tables consisted of three segments, such as shown in Figure 8 for run 1.1, has segments 1.1a, 1.1b and 1.1c. These are all about 120° apart in heading. It is NOT critical that the runs be 120°. In fact, I have used this software to reduce data from “horseshoe heading” method.

Embraer EMB 140 Cloverleaf Data The two cloverleaf maneuvers presented in this memo were performed as wind calibration maneuvers for sideslip calibration. The airspeed altitude system had been previously calibrated during certification using other accepted

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techniques. The sideslip data will not be presented here. In order to compute a sideslip angle from inertial data (velocities N, E, D and angles heading, pitch and roll), we would need true airspeed and the components of wind (N and E). The cloverleaf maneuver gives you that data. When used as an airspeed calibration maneuver, the wind components serve as a maneuver quality parameter. That is, if one were to conduct a series of runs at the same pressure altitude on the same day you would expect to see a minimal variation of wind as a function of time. This is especially true at higher altitudes. At lower altitudes, you would experience much larger variations in wind versus time. The two runs (we will label them GPS-1 and GPS-2) were on the same flight, both at an airspeed of 200 knots and pressure altitude of 15,000 feet. Figure 9 and Figure 10. are plots of the course over the ground. The differential GPS (DGPS) data was from an AshTech (subsidiary of Thales) system. [This is not meant to be an endorsement of that particular brand as the accuracies are primarily a result of the GPS constellation]. The origin (0,0) of the plots is the location of the GPS ground station. In a DGPS system, the ground station is a transmitter essentially identical to the GPS satellites. This is also called a pseudolite [a word combining pseudo and satellite]. While the satellites are some 20,000 kilometers above the earth, the ground station is less than 100 kilometers from the aircraft. This factor is what gives the DGPS, typically, accuracies down to the centimeter level versus current quoted GPS accuracies of about 3 meters. The cloverleaf method is a true airspeed method. I will compute true airspeed from the Pitot-static system data (total pressure, static pressure and total temperature). That true airspeed is denoted ( Vti ), or indicated true airspeed. The algorithm will also compute true airspeed from GPS ground speed components plus wind components. The essence is that we will have three equations (one for each of three legs of our cloverleaf maneuver) and three unknowns. The three unknowns are two components of wind (north and east) and an unknown error in true airspeed.

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GPS-1

North Distance from Ground Station (meters)

65000

60000

55000 Run 1.1

50000 Run 1.3

45000

Run 1.2

40000

START 35000

30000 50000

55000

60000

65000

70000

75000

80000

East Distance from Ground Station (meters)

Figure 9

Course over the ground for GPS-1

One minute or more of data was averaged from each of the three straight segments. The total time of this maneuver was just over 13 minutes. A suggestion is that equal quality data could be obtained in much less time. One would perform a triangle maneuver, instead of a cloverleaf. Instead of one full minute of data, with high quality DGPS data at two sps, about 10 seconds of data averaged should be sufficient. The “cloverleaf” maneuver was used on the highly maneuverable high speed pacer aircraft at the AFFTC. For slow speed aircraft, we have found the triangle maneuver to be preferable. The triangle maneuver requires a 120° turn between data points, while the cloverleaf maneuver requires a 240° turn between data points.

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GPS-2

North Distance from Ground Station (meters)

65000

60000

55000

Run 2.1

50000 Run 2.3

45000

40000

Run 2.2

35000

30000 40000

45000

50000

55000

60000

65000

70000

75000

80000

85000

East Distance From Ground Station (meters)

Figure 10

Course over ground for GPS-2

Figure 11 is a time history plot of the three legs of GPS-1. Each of the three starts at a different time, so to get them conveniently on the same plot they are all plotted versus their corresponding elapsed time. The data are the ground speeds ( Vg ), however on the plot are shown the average values for track angle ( σ g ), and indicated true airspeed ( Vti ). We will compute the error (correction to be added) in true airspeed ( ∆Vt ) as well as the components of wind speed ( VwN , VwE ).

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GPS-1 Ground Speeds [DGPS] Vg1 = 273.91 knots

280

σ g1 = 339.41° Vt1 = 258.22 knots

275

Ground Speed (knots)

270 265

Vg 2 = 252.94 knots

σ g2 = 209.81°

260

Vt 2 = 254.63 knots

255 250

Vg 3 = 242.33 knots

245

Vt 3 = 260.92 knots

σ g3 = 107.14°

240 235 0

10

20

30

40

50

60

70

80

90

Elapsed time (seconds)

Figure 11

Input data for GPS-1

The output of the Excel program contains a number of parameters. This includes all of the input parameters for each of the three legs, several airspeed calibration parameters and wind speed and wind direction. The “answer” of primary interest is the correction to be added to true airspeed. Once that number is computed, the Excel program also calculates corrections to calibrated airspeed, Mach number, pressure altitude and static pressure. These are all based upon the assumption that all of the error is due to errors in static pressure. The assumption of all the error in the Pitot-static system is due to the static pressure is the usual assumption in most air data calibrations. Results: → ∆Vt = 0.22 knots

Where the average of the three indicated true airspeeds is:

Vti = 257.92 knots The wind components were computed at the following.

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VwN = −10.68 knots VwE = 16.32 knots The key here is that we must satisfy the basic airspeed formula: G G G Vt = Vg + Vw

(1.4)

Expanding in north and east direction:

VtN = VgN + VwN and VtE = VgE + VwE

(1.5)

Then, Vt = VtN 2 + VtE 2

(1.6)

True airspeed is also the indicated true airspeed plus an error in airspeed. Setting those equal to each other yields the following. We will do the check for each of the three legs in Table 8.

Vti + ∆Vt =

Table 8

(V

+ VwN ) + (VgE + VwE ) 2

gN

2

(1.7)

Summary of inputs and outputs for GPS-1 and GPS-2

The last column is the square root term, while the fourth column on the left is indicated true airspeed plus correction term. These two runs were about 20 minutes apart. You should observe the very small corrections to true airspeed and that the computed wind speed components were nearly the same from one cloverleaf run to the next. 21

The Doug Gray Algorithm Doug Gray2 presents an Excel spreadsheet that computes true airspeed and wind speed. The primary assumption is that three airspeed vectors all lie on a circle, which is equivalent to all three airspeeds being equal. I have coded up his Excel file and reproduced the numbers in his paper. In addition, the numbers from Cessna 180 agreed exactly due to the airspeeds being identical on all three passes. When the true airspeeds are not identical, we would obtain somewhat different results. Within the assumption that each of the three passes has the same wind and error in true airspeed, our algorithm gives an exact solution.

Gray Algorithm Comparison Using Cessna 180 Data As a check case of the Cloverleaf Excel program versus Doug Gray’s Excel spreadsheet, I will use run 1.4 of the Cessna 180 as presented in Table 6 and Table 7. The one change is I will convert units to knots. I should note that the airspeed indicator can be read to at best 2 knots and here we are presenting data to nearest 0.01 knot. I am doing that to compare the algorithms – this is not to suggest that the Cessna 180 airspeed correction is good to 0.01 knots. Table 9

Comparison of Gray Algorithm with Cloverleaf

Run

Vg

σg

Vti

∆Vt

(Vt + ∆Vt )

Vw

ψw

No. 4.1a 4.1b 4.1c

Knots 91.98 85.76 90.85

Deg 145.23 26.63 261.24

Knots 91.33 91.33 91.33

Knots -1.85 -1.85 -1.85

Knots 89.48 89.48 89.48

Knots 3.82 3.82 3.82

Deg 13.40 13.40 13.40

4.1a-Gray 4.1b-Gray 4.1c-Gray

91.98 85.76 90.85

145.23 26.63 261.24

89.48 89.48 89.48

3.82 3.82 3.82

13.40 13.40 13.40

As you can see, we got identical results for the true airspeed, wind speed and wind direction. In my case, I was not solving for true airspeed but rather an error in true airspeed. The assumption is that the three unknowns are all remain constant for the three passes.

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Gray Algorithm versus F16 Data

The F16B data (Table 5) had airspeeds that were not identical on each of the three passes. We will add an additional column of corrected true airspeed ( Vt + ∆Vt ). Then, for the exact same input parameters I will compute true airspeed, wind speed and wind direction using Gray’s Excel file. We chose the second run due to it having the largest variation in true airspeed. The first three rows of Table 10 are computed using the Cloverleaf method and the last three from the Gray Excel file. Notice there is no input for true airspeed for the Gray method.

Table 10

Comparison of Cloverleaf and Gray Algorithm

Run

Vg

σg

Vt

∆Vt

(Vt + ∆Vt )

Vw

ψw

No. 2a 2b 2c

Knots 306.9 475.4 484.3

Deg 19.7 242.9 144.3

Knots 414.7 419.0 417.1

Knots -0.5 -0.5 -0.5 Average=

Knots 414.2 418.5 416.6 416.4

Knots 108.3 108.3 108.3

Deg 10.4 10.4 10.4

2a-Gray 2b-Gray 2c-Gray

306.9 475.4 484.3

19.7 242.9 144.3

416.4 416.4 416.4

110.4 110.4 110.4

11.0 11.0 11.0

As can be seen, the average true airspeeds are identical. It is only in the wind computations that we see a difference. To the nearest 0.1 knot, all of the runs evaluated in this memo had identical average true airspeed results. As in Table 10, there were small differences in winds for any run where the true airspeeds were not identical for all three passes. To compute the average true airspeed, the Gray method works great.

Finally So, I wonder why this basic concept of three speed runs to calibrate airspeed has not caught on? The ground speed course, with its basic flaw of the crosswinds, just seems to linger on. For one thing, NACA9, in their 1937 “The Measurement of Air Speed of Airplanes” downplays the method. They state that “the analytic solution is too laborious for convenience.” Granted they did not have modern computers, however, the Doug Gray algorithm should not have been “too laborious” with slide rules. That simple statement in a widely distributed NACA report, may have contributed to the end of using speed course methods or at 23

least the three leg methods. As the Germans7 demonstrated using phototheodolite cameras, one need not have a laid out course to perform a speed run method. The speed runs could be two 180 degree apart runs (“ground speed course”) or any of the 3-vector patterns. A 1995 NASA Report10 “Airdata Measurement and Calibration”, makes no mention of any speed course or speed run method. In addition, the “AFFTC Standard Airspeed Calibration Procedures11” though it has a 1981 date is simply a reprint of the original April 1968 document and it, also, does not mention any Three-Vector methods.

References 1. Olson, W.M., “Pitot-Static Calibrations Using a GPS Multi-Track Method,” Presented at SFTE Symposium, Reno, NV, 1998. [Can be downloaded at www.camasrelay.com/aircraftperformance.htm ] 2. Gray, Doug, “Using GPS to Accurately Establish True Airspeed,” June 1998. [Can be downloaded at www.ntps.edu ] 3. Lawford, J.A. and Nipress, K.R., “Calibration of Air Data Systems and Flow Direction Sensors,” pages 16-20, AGARD AG-300-1, September 1983. 4. Fox, David, “Is Your Speed True”, KITPLANES Magazine, February 1995. [Can be downloaded at www.camasrelay.com/aircraftperformance.htm ] 5. Lewis, G.V., “A Flight Test Technique Using GPS for Position Error Correction Testing,” Cockpit, Quarterly of the Society of Experimental Test Pilots, Jan-Mar, 1997, pages 20-24. 6. De France, S.J., and Burgess, C.P., “Speed and Deceleration Trials of U.S.S. Los Angeles,” NACA Report No. 318, 1930. [Report may be downloaded at www.larc.naca.gov ] 7. Heidelberg, V. and Holzel, A. “Speed Measurements Made By Division “A” of the Airplane Directorate, Subdivision for Flight Experiments,” NACA TN No. 147, July 1923. [Report may be downloaded at www.larc.naca.gov ] 8. Olson, W.M., “True Airspeed Calibration Using Three Radar Passes,” Performance and Flying Qualities Office Memo, AFFTC, August 1976. [Downloadable at www.camasrelay.com/aircraftperformance.htm ] 9. Thompson, F.L., “The Measurement of Air Speed of Airplanes,” NACA No. 616, 1937. [Report may be downloaded at www.larc.naca.gov ] 10. Haering, E.A., Jr., “Airdata Measurement and Calibration,” NASA TM 104316, December 1995. [Report may be downloaded at www.dfrc.nasa.gov ] 11. DeAnda, A.G., “AFFTC Standard Airspeed Calibration Procedures,” AFFTC-TIH-81-5. [A reprint of April 1968 document]

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List of Figures and Tables Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11

Run 1 of Ground Speed Course..........................................................4 Run 2 of Ground Speed Course..........................................................4 Graphical Solution for Airspeed and Wind speed................................5 NACA Test Technique Description .....................................................6 NACA Speed Data ..............................................................................7 Course for Airship Trials......................................................................9 Velocity Diagram from 1918 (Germany)............................................10 Latitude versus Longitude – First Run...............................................15 Course over the ground for GPS-1 ...................................................18 Course over ground for GPS-2......................................................19 Input data for GPS-1......................................................................20

Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10

NACA Data Processed With Cloverleaf Program...............................8 Airspeed Triangle Data from Germany in 1918................................11 Summary of Radar track data ..........................................................12 Summary of F-15B Pacer Cloverleaf Data .......................................13 F-16B Pacer Cloverleaf Data Summary ...........................................14 GPS Inputs for Cessna 180 .............................................................16 Cloverleaf Outputs for Cessna 180 ..................................................16 Summary of inputs and outputs for GPS-1 and GPS-2 ....................21 Comparison of Gray Algorithm with Cloverleaf ................................22 Comparison of Cloverleaf and Gray Algorithm.................................23

Abbreviations AFB – Air Force Base AFFTC – Air Force Flight Test Center AGARD – Advisory Group for Aerospace Research & Development DGPS – Differential GPS FAA -- Federal Aviation Administration FORTRAN – FORmula TRANslation GPS -- Global Positioning System INS – Inertial Navigation System NACA – National Advisory Committee for Aeronautics NASA – National Aeronautics and Space Administration SFTE - Society of Flight Test Engineers

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Symbols Note: Symbols are sorted alphabetically by description since Word does not recognize the symbols.

∆Vt - Correction to be added to true airspeed VgE - East component of ground speed VwE - East component of wind speed Vg - Ground speed Vti - Indicated true airspeed VgN - North component of ground speed VwN - North component of wind speed σ g - Track angle Vt - True airspeed ψ w - Wind direction Vw - Wind speed

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