Correction of static pressure on a research aircraft in accelerated flight using differential pressure measurements

A method is described that estimates the error in the static pressure measurement on an aircraft from differential pressure measurements on the hemispherical surface of a Rosemount model 858AJ air velocity probe mounted on a boom ahead of the aircraft. The theoretical predictions for how the pressure should vary over the surface of the hemisphere, involving an unknown sensitivity parameter, leads to a set of equations that can be solved for the unknowns – angle of attack, angle of sideslip, dynamic pressure and the error in static pressure – if the sensitivity factor can be determined. The sensitivity factor was determined on the University of Wyoming King Air research aircraft by comparisons with the error measured with a carefully designed sonde towed on connecting tubing behind the aircraft – a trailing cone – and the result was shown to have a precision of about ±10 Pa over a wide range of conditions, including various altitudes, power settings, and gear and flap extensions. Under accelerated flight conditions, geometric altitude data from a combined Global Navigation Satellite System (GNSS) and inertial measurement unit (IMU) system are used to estimate acceleration effects on the error, and the algorithm is shown to predict corrections to a precision of better than ±20 Pa under those conditions. Some limiting factors affecting the precision of static pressure measurement on a research aircraft are discussed.


Introduction
Static pressure measurement on an aircraft is inherently problematic because the pressure changes as the air accelerates around the wings and fuselage, as predicted by the Bernoulli equation.It is difficult to find a location on the aircraft to measure the true undisturbed pressure, P ∞ , i.e., the pressure at a distance far from the flow-disturbing effects.On the aircraft, static pressure P s is measured at a set of ports where the aircraft designers have determined that the error, referred to as the static defect or position error, is minimal.In addition to causing errors in pressure-derived aircraft altitude, the static defect also leads directly to errors in airspeed and other measurements that need dynamic corrections, such as temperature, for example.In the pitot tube technique of airspeed measurement (Doebelin, 1990), dynamic pressure q c = P T −P s 1/2ρU 2 is sensed, where P T is the total pressure, ρ the air density, and U the airspeed.On the University of Wyoming King Air (UWKA) research aircraft, static pressure errors can be as large as 2 % of q c at research aircraft speeds (∼ 100 m s −1 ), and this error transfers directly to an error of 1 % (1 m s −1 ) in airspeed.These errors directly affect estimates of atmospheric air motions which are sensed using the "drift method" in which the three-dimensional airspeed and ground velocity vectors are subtracted.
The static pressure ports on the UWKA are located on both sides of the fuselage near the rear of the aircraft.The ports are connected together in a manifold to compensate for the ram pressure effect that sideslipping can create on the upstream side of the fuselage.By manifolding, it is assumed that this effect will average to zero.But when lateral airspeed components occur because of turbulence, or by rudder application causing sideslipping, this assumption may not be correct, as is shown later in this paper.Further, these errors are likely to change with the deployment of wing flaps, landing gear, or the addition of external housings and fairings used to accommodate instruments.The usual approach is to develop corrections using data taken from flights past an instrumented tower, or from precise static pressure sources towed behind the aircraft (Brown, 1988;Wendisch and Brenguier, 2013).
Static pressure becomes more problematic when aircraft are used in the study of pressure fields in baroclinic zones or cloud systems.Bellamy (1945) introduced airborne determination of D-value, the difference between the radardetermined geometric altitude and the pressure altitude from static pressure, using the standard atmosphere assumption.Shapiro and Kennedy (1981) and Brown et al. (1981) applied this to the determination of jet stream geostrophic and ageostrophic winds.This pressure gradient approach has also been used for studies of low-level jets (Rodi and Parish, 1988;Parish et al., 1988;Parish, 2000).LeMone and Tarleton (1986) and LeMone et al. (1988) used altitude derived from accelerometer measurements instead of radar altitude in perturbation pressure studies around clouds suggesting that accuracies of 20 Pa can be obtained, but only with substantial empirical corrections and carefully flown legs.More recently, Parish et al. (2007) and Parish and Leon (2012) demonstrated that Global Navigation Satellite System (GNSS, hereafter referred to as global positioning system -GPS) data can resolve both pressure gradients and perturbations associated with mesoscale and cloud-scale systems.
In this study, we develop and test a method for estimating static defect using differential pressure measurements on the hemispherical leading surface of a Rosemount model 858AJ (hereafter R858) air velocity probe.The theoretical predictions for how the pressure should vary (presented in the Appendix) lead to a set of equations that can be used to solve the static pressure error in addition to the attack angle, sideslip angle and dynamic pressure.We first use trailing sonde data to determine the probe sensitivity factor, and then compare resulting error estimates with accurate altitude measurements from a GPS-aided inertial measurement unit (IMU), allowing for an independent check of the precision of the algorithm and an examination of the effects of aircraft acceleration.
2 Retrieval of static pressure error from differential pressure measurements Bögel and Baumann (1991) describe a method of analysis of R858 measurements during pilot-induced maneuvers to estimate static pressure errors.Crawford and Dobosy (1992) describe the Best Aircraft Turbulence (BAT) differential pressure flow-angle probe which addresses the static pressure problem by averaging pressure on several ports.Here, we develop a method to predict the static pressure error directly by using pressure measurements from the R858 air velocity probe which, on the UWKA, is mounted at the tip of a boom, as shown in Fig. 1.The probe is a hemispherecylinder configuration in which a hemispherical surface is at the end of a 2.5 cm diameter, 12.5 cm long cylindrical section.Pressure measurements on ports in a hemispherical surface are used for airspeed and flow angle determination (Brown et al., 1983).There are 5 ports: one central port which approximates total pressure, two ports separated by ±45 • from the central axis in the aircraft horizontal plane for sideslip angle determination, and two ports separated by ±45 • in the aircraft vertical plane for attack angle determination.In the UWKA system, four differential pressure measurements are made, allowing the static error to be estimated.The method and specific equations used in the UWKA R858 configuration are presented in the Appendix.The attack angle α, for example, can be estimated as described in the manufacturer's technical note (Rosemount, 1976) using where the numerator is the differential pressure between the two attack angle ports.The sensitivity coefficient, assuming potential flow (using a sensitivity factor, as defined in the Appendix, f = 9/4) for small angles, is K ∼ = 2f (π/180) = 0.0785 deg −1 .Brown et al. (1983) investigated K using data from flight maneuvers for the R858 on the noseboom on the NCAR Sabreliner.They found that for N Mach < 0.5, K = 0.068 deg −1 (f = 1.95), 13 % lower than the value recommended in Rosemount (1976).This determination of K was not exact nor without ambiguities, however.The Brown et al. (1983) method substituted precise IMU-measured pitch angle for attack angle, which is valid during periods of straight and level flight assuming zero vertical wind and aircraft vertical velocity.Using pitch in this manner, the upwash effect (Crawford et al., 1996) of the fuselage, wings, and possibly the R858 itself is incorporated into the value of K.The sense of upwash effect is to make the local attack angle larger than the pitch angle, causing K to be overestimated compared to the local value of K without the upwash effect.Brown et al. (1983) apparently also used P 1 for q without static defect or flow angle correction, which could further bias the Kvalues reported.In the present study, the sensitivity factor is estimated directly so that the upwash effect is not accounted for so that the flow angles that result are relative to the hemispherical surface, and q is corrected for flow angle.
We now show that the static pressure error can be determined from the differential pressure measurements, assuming that sensitivity factor f can be determined.The theoretical predictions of the four differential measurements from the Appendix (Eq.A11) are: where here we have replaced P ∞ in Eq. ( A11) with the measured static pressure P s,m less the unknown error P err , and the remaining unknowns are the attack angle α, sideslip angle β, dynamic pressure q, and the probe sensitivity f .We note that we are solving for the error, using the expression for P 1 in Eq. ( 2) rearranged as We show in the Appendix that α and β can be found without a priori knowledge of f .Thus, P err is the departure of P 1 from q after the attack and sideslip angle correction.The strategy is to develop an estimate f using the trailing cone test data, described in the next section, and then estimate P err as one of the unknowns from the four pressure measurements.3 Trailing cone test data Ikhtiari and Marth (1964) and Mabry and Brumby (1968) describe an aircraft pressure calibration system in which a static pressure probe, connected to the aircraft by tubing, is towed at some distance behind and below the aircraft, away from the disturbing influence of the aircraft.The towed sonde is stabilized in flight by means of a carefully engineered cone -thus the term "trailing cone" (hereafter called TC ) -that creates drag for stabilization of static pressure ports which are distributed around the circumference of a straight piece of tubing behind the cone.Brown (1988) described the technique in detail, and demonstrated for the NCAR Sabreliner that the largest expected error in the pressure measurement is ±39 Pa after application of corrections based upon the TC data.The distance that the TC is extended is chosen to be long enough to minimize pressure fluctuations in the aircraft's wake.However, accelerations cause errors.For example, with a 19 m tubing length (the length used for the UWKA), a 0.1 g acceleration will result in a 15 Pa effect.Consequently, we limit data collection to steady, straight flight legs.
A UWKA test flight using the Douglas model 501 TC was conducted on 27 October 2005.Data were collected in several configurations: (1) with the aircraft in "clean" configuration (landing gear raised, flaps retracted), (2) in segments with gear lowered, (3) with gear lowered and flaps extended in approach mode, and (4) at different power settings and altitudes.Several measured variables are shown in Fig. 2 as a function of aircraft attack angle, using data filtered at 5 Hz and output at 10 Hz.The lack of a single relationship for all configurations among dynamic pressure, pressure error, and N Mach is evident.This is the main obstacle to simple corrections as a function of one variable such as dynamic pressure or attack angle.

Empirical determination of f from trailing cone data
We first solved Eq. ( 2) for the unknown values of f , α, β, and q c using TC flight measurements of P err and differential pressures P 1 , P α , P β , and P R .The resulting fvalues are plotted in Fig. 3, and show increases with P α , and decreases with N Mach .We then used a non-linear regression procedure to obtain the empirical estimate of f that best predicts the TC-determined P err .The best fit was found by trial and error selection of variables to be: where the constants were found to be c 0 = 1.700, c 1 = −0.1569,c 2 = 0.06633 and c 3 = +0.001254,and where P α is in units of hPa.A discussion of the physical basis for these relationships will be presented later.
The resulting predictions of pressure error are compared with TC measurements in Fig. 4 at various steady flight configurations and power settings.The distribution of the residual errors is shown in Fig. 5 to have a precision ±10 Pa in all configurations (σ = 8 Pa).
There is additional evidence for the variability of f .Brown et al. (1983) and Rosemount (1976) provide experimental evidence for f decreasing with N Mach , consistent with the present results.Traub and Rediniotis (2003) (TR03) present an analytical prediction of surface pressures for a hemisphere-cylinder configuration similar to the R858, and wind tunnel results at Reynolds numbers about a factor of two higher than UWKA flight (N Re ≈ 1.5 × 10 4 ).The TR03 theoretical formulation, confirmed by their wind tunnel results, predicts sensitivity to be f = 2.07 at zero incidence angle, and also their data show that the sensitivity varies with incidence angle.
To explore the effect of probe shape on f , we used a commercially-available finite-element solver for turbulent, compressible flow equations (FIDAP, Fluid Dynamics International, Inc.).An axisymmetric, compressible model at zero attack angle was used and showed that f is different for various mounting configurations, and changes with speed.Solutions were found for N Mach equivalent to speeds of 50-180 m s −1 .Four configurations were modeled as shown in Fig. 6 (not to scale): (1) a sphere (not shown); (2) the R858 probe mounted on the end of the UWKA nose boom, which is 0.2 m in diameter; (3) the spherical radome (nose radar covering) as on the former National Science Foundation (NSF) King Air aircraft (N308D); and (4) the R858 mounted on a nose boom four times the diameter of the actual UWKA nose boom.Pressure distributions on the spherical surface were fit using the sin 2 relationship (A1), and the resulting f for each N Mach plotted in Fig. 7a.For the sphere, f varies from 1.9-2.1.Adding the nose boom behind the R858 hemisphere-cylinder decreases to about 1.9.Increasing the boom diameter to 0.8 m lowers f to about 1.5.These results suggest that there is no unique value of f for all mounting configurations of the R858.Indeed, the fuselage of the aircraft itself presents a formidable aerodynamic barrier to the probe, which is likely to contribute to the actual f variability in addition to a particular mounting configuration.
To explore further the behavior of f , we constructed a physical model of the R858 with extra pressure ports drilled so that adequacy of the sin 2 relationship between angle and pressure could be determined by direct measurement along with determination of f .The test was conducted in the University of Wyoming Low Speed Wind Tunnel, which has a test section of 0.6×0.6×0.9 m.The model was constructed 75 % larger than the actual probe so that the Reynolds number at the test speed of 50 m s −1 would be approximately that of the actual (0.0254 m diameter) probe at 90 m s −1 , the typical flight speed for the UWKA.Data were digitized with a personal-computer-based data logging system at 10 Hz after analog filtering with cutoff frequency of 2 Hz.The analysis then was performed on one minute averages of pressures at each port.Additional measurements made with a standard pitot tube placed upwind from the test body to ensure that the tunnel speed did not change during runs at each attack angle.In Fig. 7b, the sensitivity of f to N mach from Brown et al. (1983) and (Rosemount, 1976) are shown, along with f determined from the wind tunnel tests (point labeled as "T "), and the solution of Eq. ( 2) with actual King Air flight data.

Measured pressure compared to pressure derived from GPS altitude
While the 2005 TC test flight varied attack angle with airspeed and altitude, sideslip angles were intentionally kept near zero to prevent lateral acceleration.In this section, the efficacy of the estimates of P err in an accelerating flight environment using the differential pressure solution described in the previous section will be examined.Differential global positioning system (dGPS) techniques use data from one or more stationary reference or base GPS stations which have precisely determined location to refine position estimates for the receiver on the aircraft.dGPS processing techniques using dual-frequency (L1/L2) carrier phase data can eliminate errors caused by ionospheric and tropospheric delays entirely, resulting in position estimates with accuracy of centimeters under optimal conditions (Parish et al., 2007).
In the present study, we use post-processed inertial measurement unit (IMU) data in conjunction with dual-frequency dGPS data to resolve the aircraft position and motion  The accuracy includes effects of non-linearity, repeatability, temperature (−51 to 80 • C) and calibration.Worst case error from transducer acceleration is specified to be 20 Pa under acceleration of 6 g; the maximum acceleration in these tests was ±1 g.We estimate the maximum dynamic errors in the connecting tubing to be 10 Pa for longitudinal accelerations (here 0.1 g, 10 m tubing length) and lateral (1 g, 1 m tubing length) accelerations of the air column.
Flight data were collected on 16 September 2011 during pilot-induced maneuvers inducing variations in attack and sideslip angles.Periods of turns were not considered in the analysis.The aircraft was flown at nominally constant pressure with deviations corrected hydrostatically to that pressure using the method described by Parish et al. (2007).Pressurederived altitude changes were determined from integration of the hydrostatic equation:  z − z 0 = − where z 0 is a direct measurement of geometric altitude from the IMU/dGPS system at pressure P 0 and virtual temperature T v .Data were collected at two altitudes -nominally 4600 and 7000 m m.s.l.Other relevant measurements include in-house developed reverse flow temperature (accuracy of 0.5 K, resolution of 0.006 K), and Edgetech Model 137 dew point temperature (accuracy of 1 K, resolution 0.006 K).
A bias is introduced if an atmospheric horizontal pressure gradient exists, or when pressure is falling, along the flight track since constant geometric height is no longer constant pressure.To minimize this effect, the time series is broken up into 500 s segments and reinitialized with z 0 from the highly accurate IMU/dGPS geometric altitude value at that instant.The Laramie Valley was under the influence of high pressure, clear sky, and weak pressure gradient during the analysis period, which minimized the pressure change effect.
Computing geometric altitude from measured static pressure also involved carefully considering the relative distance vectors of the inertial measurement unit (IMU), GPS antenna, R858, and the static pressure locations.These vectors were accurately determined with accuracy < 5 cm using precise surveying techniques.The resulting "lever arm" fluctuations were less than 8 Pa during the sideslip and 15 Pa during the attack angle changes, as shown in Fig. 8d.
The angle changes with periods of about 10 s, as shown in Fig. 8b, result in correlated vertical accelerations as shown in Fig. 9.The attack angle was limited to −2 to −12 • , resulting in ±1 g changes, helping to avoid excessively large altitude excursions.The sideslip angles (Fig. 8c) were restricted to

Frequency response
The power spectral density (PSD) plots of 100 Hz static pressure and correction data are shown in Figs. 13 and 14 for attack angle and sideslip angle changes, respectively.Data is from the 7000 m altitude in which the atmospheric turbulence is very low.Also shown on the plots is the vertical wind speed in these legs.The derived pressure correction from the R858 algorithm is almost 2 decades in power below the uncorrected pressure, indicating that the correction has minimal adverse effect.Also noted is that the pressure shows whitening at frequencies higher than 1 Hz.This noise is probably aerodynamic in origin with a standard deviation

Summary and discussion
An algorithm was developed to estimate static pressure errors in steady flight using R858 differential pressure measurements, and tested with trailing cone data from a University of Wyoming King Air flight in 2005.After calibrating f using the TC flight data, it was shown that the effects of speed, altitude and flight configuration (landing gear, flap extension) can be predicted to σ = 8 Pa in steady flight.To capture the effects of acceleration, flight maneuvers were conducted and geometric altitude from a GPS-aided inertial measurement data were used to predict the pressure error.These results suggest that pressure errors can be determined with a precision of ±20 Pa during such maneuvers.It should be emphasized that the precision, not the absolute accuracy, of these estimates is addressed in this paper.The absolute accuracy of the error estimates using this method depend on this empirical determination of f as well as other factors addressed in the present work.
Our attempts to use statistical regression analysis alone to relate the observed pressure error to flight data (N Mach , q c , inertial acceleration) have not been successful.The differential pressure method, however, has the advantage of being a solution based upon the R858 equations, as presented in the Appendix.The main weakness is that determination of the probe sensitivity f requires an independent means of its determination.In the present study, we used the TC measurements to calibrate f .
There are several sources of error which may be a factor in the interpretation.The effect of acceleration of the air in the connecting tubing, which we estimate to be smaller than 10 Pa, is indistinguishable from the aerodynamic cause of the static defect at the sensing ports.Nonetheless, the R858 pressure imbalance approach should capture the connecting tubing effect.The other factor is the error introduced by uncertainty in height differences between the static pressure ports at the rear of the fuselage, IMU and GPS antenna locations, and R858 probe tip.Figure 8d shows this effect is < −15 Pa.
Our flow modeling suggests that the relatively low estimates of f from the algorithm (f ∼ = 1.7) may be reasonable since f was found to decrease as the structure behind the R858 hemisphere-cylinder gets larger (Fig. 7).These values are lower than TR03 (hemisphere-cylinder, f = 2.07) which also suggest that f varies with incidence angle.It would be useful to obtain independent confirmation of these estimates, for example, by using pitch angle as a surrogate for α at differing airspeeds, as described by Brown et al. (1983).But that approach has limitations, as discussed in Sect.2, especially since using pitch as an estimate of α implicitly incorporates the upwash effect (Crawford et al., 1996) into the sensitivity.Further, using a separate pitot tube measurement for q would itself require the correction for static defect.The problems extend to the horizontal with sidewash effects.Thus, we think that independently estimating f , while desirable, is problematic at best and beyond the scope of this paper.
One possible shortcoming of the theoretical prediction of pressure distribution on the hemisphere, as shown in the Appendix, is the a priori assumption that f is constant, while TR3 suggests that f varies with attitude angle.Further, the flow modeling suggests that f may be different for the vertical and lateral axes when the probe is ahead of an asymmetrical body like an aircraft fuselage and wing.The complete characterization of the probe would require a 3-dimensional flow modeling of the entire aircraft with the boom and probe.

Fig. 2 .
Fig. 2. Scatter plots (10 Hz data) as functions of attack angle [deg] for various flight configurations and altitudes as indicated in legends: (a) dynamic pressure q c [hPa], (b) and (c) static pressure error [Pa] determined by TC, and (d) Mach number.

Fig. 3 .
Fig. 3. Scatter plots (10 Hz data) of probe sensitivity factor f , calculated from the solution of Eq. (2), versus P α [hPa] (top panel), and Mach number (bottom panel) for various flight configurations and altitudes, as indicated in legend.

Fig. 4 .
Fig. 4. Pressure error [Pa] predicted from differential pressure algorithm vs. error measured with TC for various flight configurations, altitudes and propellor speeds, as indicated in legends (10 Hz data). 1 : 1 lines are shown.

Fig. 7 .
Fig. 7. Estimation of sensitivity factor f from (a) modeling and (b) values from Brown et al. (1983), (Rosemount, 1976), and solution to Eq. (A11) for UW King Air flight data.The point labeled "T " represents the value from determined from the wind tunnel tests of the R858 model.

Fig. 8 .
Fig. 8. Time series of (a) height difference (geometric-pressure) [m], (b) attack angle [deg], (c) sideslip angle [deg], and (d) pressure deviation from lever arm correction [Pa].Time series is the concatenation of four 500 s segments, the first two flown at 4600 m, the second two at 7000 m.

Table 1 .
Trimble/Applanix airborne positioning system perforh −1 ) * 0.5 * Attitude will drift at this rate up to the maximum absolute accuracy.
±8 • , limiting lateral accelerations to ±0.20 g lateral accelerations for crew comfort and also safety considerations.Typical values during research flights are less than ±0.5 g, ±1 g in strong turbulence, and only rarely experiencing the ±2 g limitation set under U.S. Federal Aviation Administration Part 91 certification.In our experience, instantaneous attack angle values in severe turbulence have been noted to reach 20 • while the aircraft is still within the g-loading limits, but this is rare.The resulting static pressure errors from the maneuvers are shown in Fig.10.The pressure-geometric altitude errors for sideslipping have a range of about 50 Pa and for attack angle changes a range of about 300 PaThe errors before and after correction are shown in Fig.11for the attack and sideslip changes, with the distribution of errors before and after correction shown in Fig.12.After correction, the biases in during the attack and sideslip changes were −14 and +10 Pa, respectively, with standard deviations of 16 and 11 Pa, respectively.

Fig. 11 .
Fig. 11.Time series of errors [Pa] from Fig. 10: (a) and (b) are before and after correction, respectively, for attack changes; (c) and (d) before and after correction, respectively, for sideslip changes.

Fig. 12 .
Fig. 12. Distribution of errors [Pa] for data shown in Fig. 11 before correction (red) and residual after correction (blue) for (a) attack angle changes; and (b) sideslip angle changes.

Fig. 14 .
Fig. 14.Power spectral density as in Fig. 13 for period of sideslip angle changes at 7000 m altitude.