Kitabı oku: «Introduction to Flight Testing», sayfa 7

3.3.2 Accelerometers

A three‐axis accelerometer provides a direct measurement of acceleration in all three directions, with the acceleration being relative to the local inertial frame of reference. Thus, an aircraft in steady level flight will have an acceleration of 9.81 m/s² in the vertical direction (+1g), and zero acceleration in the other directions, since the local inertial frame corresponds with a freely falling object and the aircraft is accelerating upwards relative to the inertial frame. Accelerometers essentially operate as a spring‐mass‐damper system, where the applied acceleration along a particular axis will cause a displacement of the mass by an amount that depends on the properties of the system. The displacement is then measured by converting it to a voltage, and calibrating this voltage to acceleration. In MEMS devices, the spring‐mass‐damper system is often a cantilevered beam with a proof mass, where transduction of the displacement to voltage is most often done through capacitive or piezoresisitve schemes. Capacitive transduction involves a gap between the cantilevered beam and a fixed beam, which varies the capacitance of a circuit which can then modulate a measured voltage. The piezoresistive scheme involves a piezoelectric material as the spring in the system, where the voltage drop across the piezoresistor changes with the applied strain due to acceleration.

Accelerometers can be used in flight testing to detect events with sudden changes in acceleration (such as stall), measure the orientation of the aircraft relative to the ground in steady flight, indicate the bank angle in a steady turn by measuring g's in the aircraft's frame of reference, determine the period of dynamic stability phenomena such as the long‐period phugoid mode, and other applications. In principle, the measured acceleration can be integrated in order to infer a change in velocity, although the noise in the signal often precludes this in practice. Further, the measured accelerations are integrated into the flight data computer for determining vehicle state in modern avionics systems.

3.3.3 Gyroscopes

Building upon our earlier discussion of gyroscopic principles for traditional flight instruments, we will now consider how they are used for modern avionics systems and instrumentation. Rate gyros used in modern glass panel avionics and DAQ systems are based on MEMS‐fabricated gyros. These gyros sense the rate of angular motion, rather than directly measuring the angle itself. A three‐axis gyro will have three independent rate gyros mounted along mutually perpendicular planes. One example of a MEMS rate gyro is based on the principle of sensing Coriolis acceleration. In this configuration, a proof mass is mounted on springs and oscillated in a direction perpendicular to the measured axis of rotation. As the proof mass oscillates, its radial distance from the aircraft's center of rotation also changes, leading to time‐varying tangential velocity that subjects the mass to varying amounts of Coriolis acceleration while the body rotates. This leads to a time‐varying reaction force in a direction perpendicular to the axis of rotation and the direction of oscillation of the proof mass. This reaction force is applied across springs in the lateral direction, which translate the reaction force into a linear motion that is sensed by capacitive elements (interdigitated fingers).

Since rate gyros measure the angular speed – pitch rate, roll rate, or yaw rate – rather than directly measuring the respective angles, the rotation rates must be integrated with respect to time in order to determine the pitch, roll, or yaw angles. Through the integration process, any noise present in the signals accumulates and leads to growing error in time.

3.3.4 Magnetometers

A three‐axis magnetometer can be used to sense the aircraft's magnetic heading, much in the same way that a compass provides magnetic heading by always pointing toward magnetic north. On a MEMS‐based device, each of the three magnetometers is mounted mutually perpendicular with a reasonably high degree of precision. Each magnetometer senses the local magnetic field via the Hall effect, whereby a voltage difference is induced across an electrical conductor in a direction transverse to both an applied magnetic field and an electric current flowing through the device. For sensing the aircraft's heading, each axis of the magnetometer responds to Earth's magnetic field lines, which are aligned between the magnetic North and South poles. Since the North and South poles are not aligned with the physical poles, there is a difference between magnetic north and true north (referred to as magnetic declination, or variation). The amount of magnetic variation depends on geographic location and time, with a time scale of years (National Centers for Environmental Information 2019).

When using a magnetometer in an aircraft, the sensing of the Earth's magnetic field can be strongly influenced by nearby magnetic fields and ferromagnetic materials, through an effect known as magnetic deviation. Thus, an aircraft's compass must be calibrated to account for these error sources – this is typically done by positioning the aircraft along various points of the compass (typically referenced to a so‐called “compass rose” painted on the ramp with 30° heading intervals) and noting the difference between the compass reading and the actual magnetic bearing. The two error sources are referred to as hard and soft iron distortions. Hard iron distortions are due to the presence of nearby magnetic fields, which could be from magnets or electronic circuits, and result in a constant bias offset on the measured magnetic field. Soft iron distortions are due to the presence of nearby ferromagnetic materials (such as the engine block), and result in skewing of the measurement or an offset whose magnitude is heading‐dependent. These error sources can be nontrivial when using a three‐axis magnetometer on board an aircraft. Thus, a magnetometer that is part of an external DAQ system should be calibrated before flight to remove these effects (the magnetometers built into the aircraft's built‐in avionics have already been calibrated).

Fortunately, a compass rose is not required for calibration of a MEMS magnetometer, and the required calibration process is fairly straightforward. Before takeoff, with all electronics operating, the DAQ unit in place, and the magnetometer acquiring data, the aircraft should be swung through two complete 360° circles (a total heading change of 720°). When plotted in time, data from the x‐ and y ‐ axes of the magnetometer from this maneuver will form an ellipse. If the magnetometer was not subject to any error sources, the acquired data should form a constant radius circle centered on the origin. However, the error sources result in an elliptical pattern, where the offset of the ellipse from the origin (0, 0) depends on the error from hard iron distortions, and the eccentricity of the ellipse depends on the magnitude of error from soft iron distortions. A mapping can be determined to take the acquired data and transform it to a circle. This can be done by the following equation,

(3.3)

where x_nc, y_nc, and z_nc are the noncalibrated data straight from the magnetometer; the B vector contains the bias errors due to hard iron distortions, the M matrix corrects for the soft iron distortions; and x_c, y_c, and z_c are the calibrated data. Values of M and B are determined from a least‐squares fit to the data and subsequently applied to all magnetometer data acquired during the flight test.

3.3.5 Barometer

A MEMS pressure transducer can be used as a barometer for an indication of pressure altitude. MEMS barometers are typically piezoresistive devices, where the deformation of a thin silicon diaphragm is measured through piezoresitive principles. The silicon diaphragm is doped at certain locations, thus locally altering the electrical conductivity and imparting resistor‐like properties. As the diaphragm deflects in response to an imposed pressure, the resistances of the doped regions change. If these piezoresistors are connected to a Wheatsone bridge, the change in resistance is converted to a change in voltage that can be easily measured. The calibration of these sensors is typically temperature dependent, so MEMS barometers often include an integrated temperature sensor for compensation.

The pressure reading of a MEMS barometer can be converted to altitude via hydrostatics. Necessary input values are the local barometric pressure reading and/or field elevation, along with OAT. All of the necessary theory for this conversion is detailed in Chapter 2 on the definition of the standard atmosphere. The pressure reading (and indicated altitude) of MEMS barometers used in some external DAQ devices can have a nonnegligible bias error (offset) in the reading. This is a minor concern, however, since it is a straightforward matter to determine the offset when the aircraft is on the ground, where a pressure measurement is made and field elevation and/or local barometric pressure are known. For flight testing, the other predominant source of error is a difference between cabin pressure (where the DAQ unit is typically mounted for student flight testing) and freestream static pressure. Gregory and McCrink (2016) evaluated the magnitude of this error for flight in a Diamond DA40 and found that cabin pressure was approximately 140 Pa higher than freestream static pressure at cruise conditions. For this particular aircraft, ram air effects that slightly pressurize the aircraft cabin in flight are the likely cause of the discrepancy. Despite these error sources, altitude indication from cabin pressure tends to be more accurate than GPS‐reported altitude.

3.3.6 Fusion of Sensor Data Streams

As noted in the previous subsections, each individual sensor has some limitations to its utility due to various sources of noise or other errors such as temperature drift. This makes direct inference of vehicle state from one set of sensors problematic (e.g., integrating the signals from the rate gyros alone to determine vehicle orientation would introduce substantial errors). Optimal performance, however, is obtained when different data streams can be fused together in a manner such that the collected set provides a much more accurate and reliable estimate of vehicle state. One very common method for fusing together data streams is Kalman filtering. The Kalman filter is an algorithm that takes multiple time records of sensor data and produces estimates of vehicle state with improved accuracy compared to the estimate if only a single data stream were available. In essence, it gives a solution or prediction of vehicle state with an accuracy that is much improved over the estimate provided by any subset of sensors. The basis of the Kalman filter is described as follows.

First, we need a precise definition of the vehicle's state. It is the set of data that completely describes the vehicle's position and orientation, along with the rates of change of position and orientation. Thus, the position estimate and its derivatives would be the spatial location of the vehicle, its velocity, and its acceleration in some frame of reference. The orientation estimate is the vehicle's pitch, yaw, and roll angles, along with rates and accelerations of those same angles. These estimates of position and orientation have some uncertainty associated with them, which the Kalman filter assumes to be random and Gaussian distributed.

At each time step, a prediction of the vehicle's state at the next time step is formulated based on knowledge of the current state, an estimate of the uncertainty of the knowledge of that state, and a model based on the vehicle dynamics. When that next time step arrives, the prediction of the vehicle state is compared with measured data of the vehicle's state (along with the associated uncertainty in both the prediction and the measurements of the state). A refined estimate of the vehicle's state is generated by a weighted sum of the predicted state and measured state. Since the Kalman filter relies on both measurements of the vehicle state and model predictions of the vehicle state, the accuracy of the resulting state estimate is dramatically improved.

This is just a very brief overview of data fusion using Kalman filtering, for the purposes of providing perspective. These filters are routinely used in cockpit avionics systems, unmanned aerial vehicle (UAV) flight control systems, and in DAQ systems to improve the reliability and accuracy of vehicle state estimation. A much more detailed discussion of Kalman filtering is available in the literature (e.g., Rogers 2007; Zarchan and Musoff 2015).

3.4 Summary

We have covered a broad range of content related to instrumentation in this chapter. The central theme of this chapter is a description of how various aspects of aircraft performance can be measured in flight, and the key operating principles of standard aircraft instrumentation and avionics, as well as dedicated flight test instrumentation.

Standard aircraft instruments that are commonly used in flight testing include the airspeed indicator, altimeter, vertical speed indicator, engine tachometer and manifold pressure, and the heading indicator. MEMS‐based sensors include GPS, magnetometers, accelerometers, gyroscopes, and barometer. The general utility of these sensors across the range of flight tests presented in this text is mapped out in Table 3.1, assuming that raw data from each sensor are used individually (rather than the robust estimation of vehicle state via Kalman filtering, which may likely be beyond the scope of an undergraduate course on flight testing). Table 3.1 clearly shows that GPS, accelerometers, and the barometer are the most broadly useful sensors. Each particular sensor has strengths and weaknesses, as discussed earlier in this chapter, making them more or less useful for acquiring flight test data. This concise summary of the relevancy of each sensor provides an overview of the relative importance of each sensor and a quick reference for test planning for a given flight test. Specific details for various performance flight tests are covered in detail in Chapters 7–16.

Table 3.1 DAQ sensor utility matrix.

GPS provides ground speed, heading, and location at a maximum sample rate of about 1 Hz. Altimetry provided by GPS can have substantial errors: the vertical dimension is the least accurate for GPS positioning due to the geometry of the position estimation problem, the fact that antenna reception may be poor from within an aircraft (particularly for a smartphone‐based GPS receiver), and because the position estimation algorithms are optimized for terrestrial applications. Also, measurements of distance with GPS must involve transformation of latitude/longitude coordinates into an appropriate local Cartesian plane before further analysis can be done. Location, ground speed, and heading information are useful for nearly every flight test described in this text.

The magnetometer, in practice, is of marginal utility for aircraft performance flight testing. This is primarily because heading information is also available from the GPS receiver, the fact that the magnetometer signal is relatively noisy, and calibration is required. The magnetometer indication must be calibrated for hard and soft iron sources via a quick pre‐flight maneuver consisting of two complete turns on the ground. Postprocessing of this calibration and application to the magnetometer data for derivation of heading are more involved. However, the magnetometer provides heading information at a more rapid rate than GPS, and it is highly advantageous to incorporate magnetometer data into a state estimation algorithm such as the Kalman filter.

Three‐axis gyroscopes are also of marginal utility as a stand‐alone sensor, since MEMS gyros measure angular rates instead of the absolute angles that are more relevant for determination of aircraft attitude. Further, significant noise can be introduced when integrating these signals in order to determine aircraft attitude.

Accelerometers provide a good indicator of any transient event found in flight testing, making it straightforward to find that event in a long data record. Features in the flight such as the stall event, takeoff point, measurement of load factor in a turn, and identification of the characteristic frequency in dynamic longitudinal stability are relatively straightforward to find and measure in accelerometer data. The accelerometers are also useful for measuring the frequency of engine vibrations, which is an indicator of engine speed (rpm), as long as the sampling rate is high enough to avoid aliasing (discussed in the next chapter). Accelerometer signals also facilitate identification of dynamic stability characteristics of the aircraft, such as the phugoid or Dutch roll modes. One key limitation of a 3‐axis accelerometer is that the DAQ device axes may not be coaligned with the aircraft body axes, but this can be calibrated by comparing accelerometer signals during steady (1 g) flight.

A barometer can provide a measurement of cabin pressure, which serves as a reasonable proxy for freestream static pressure in most situations. Both altitude and vertical speed can be reliably estimated from barometry data. Pressure altitude can be inferred using the local barometric pressure reading and refined with a measurement of outside static temperature.

Finally, we will conclude with a note of caution about comparing data from different sensors to infer flight test results. Some sensors can have inherent temporal delays, which could lead to timing or phase mismatch between sensor streams. For example, the computation of GPS position estimates requires processing time on board the receiver, so it is important to make sure the correct time base is used for comparison of GPS data with other sensor data streams. Similarly, different sensors (even those installed on the same DAQ device) can have different sample rates. For example, GPS data are often sampled at a rate on the order of 4 Hz, while accelerometers may be sampled at 100 Hz. Again, it is important to use the correct time base for comparing data streams. The next chapter will discuss in detail how data streams are digitized, along with analysis techniques such as filtering and spectral analysis.

Nomenclature

B

magnetometer calibration, bias error vector

g

gravitational acceleration

I

moment of inertia

k

recovery factor

L

angular momentum vector

M

magnetometer calibration matrix

M_ℓ

local Mach number

M_∞

freestream Mach number

t

time

T_∞

freestream static temperature

T₀

stagnation temperature

y_c

calibrated magnetometer data,

y

‐component

z_c

calibrated magnetometer data,

z

‐component

x_nc

uncalibrated magnetometer data,

x

‐component

y_nc

uncalibrated magnetometer data,

y

‐component

z_nc

uncalibrated magnetometer data,

z

‐component

γ

ratio of specific heats

ω

angular velocity vector

Subscripts

c

calibrated

nc

uncalibrated

Acronyms and Abbreviations

ADC

air data computer

AHRS

attitude and heading reference system

ATC

air traffic control

AWOS

automated weather observing system

GLONASS

Globalnaya Navigazionnaya Sputnikovaya Sistema (Russian GNSS)

GNSS

global navigation satellite system

GPS

Global Positioning System

IMU

inertial measurement unit

MEMS

microelectromechanical systems

MFD

multifunction flight display

MSL

mean sea level

OAT

outside air temperature

PFD

primary flight display

RTK

real‐time kinematic

UAV

unmanned aerial vehicle

WAAS

Wide Area Augmentation System

WGS84

World Geodetic System 1984 ellipsoid model

References

Gracey, W. (1980). Measurement of Aircraft Speed and Altitude. NASA‐RP‐1046, https://ntrs.nasa.gov/citations/19800015804.

Gregory, J.W. and McCrink, M.H. (2016). Accuracy of smartphone‐based barometry for altitude determination in aircraft flight testing. AIAA 2016‐0270, Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, CA.

Kaplan, E.D. and Hegarty, C. (2017). Understanding GPS/GNSS: Principles and Applications, 3e. Boston, MA: Artech House.

Misra, P. and Enge, P. (2010). Global Positioning System: Signals, Measurements, and Performance, 2e. Lincoln, MA: Ganga–Jamuna Press.

National Centers for Environmental Information, and British Geological Survey (2019). World magnetic model. In: National Ocean and Atmospheric Administration. http://ngdc.noaa.gov/geomag/WMM/.

National Imagery and Mapping Agency. (2000). Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems. NIMA Technical Report TR8350.2, 3e, Amendment 1, http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf.

Pavlis, N.K., Holmes, S.A., Kenyon, S.C., and Factor, J.K. (2012). The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). Journal of Geophysical Research, vol. 117 (B04406): 1–38. https://doi.org/10.1029/2011JB008916.

Rogers, R.M. (2007). Applied Mathematics in Integrated Navigation Systems, 3e. Reston, VA: American Institute of Aeronautics and Astronautics.

Sabatini, R. and Palmerini, G.B. (2008). Differential global positioning system (DGPS) for flight testing. In: RTO AGARDograph 160, Flight Test Instrumentation Series, vol. 21.

Snyder, J. P. (1987). Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1532, http://pubs.er.usgs.gov/publication/pp1395.

Titterton, D.H. and Weston, J.L. (2004). Strapdown Inertial Navigation Technology, 2e. Stevenage, Hertfordshire, UK: The Institution of Electrical Engineers https://doi.org/10.1049/PBRA017E.

Wasmeier, P. (2015). Geodetic Transformations Toolbox. Matlab Central, http://www.mathworks.com/matlabcentral/fileexchange/9696-geodetic-transformations-toolbox.

Zarchan, P. and Musoff, H. (2015). Fundamentals of Kalman Filtering: A Practical Approach, 4e. Reston, VA: American Institute of Aeronautics and Astronautics.