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
Mℓ
local Mach number
M∞
freestream Mach number
t
time
T∞
freestream static temperature
T0
stagnation temperature
yc
calibrated magnetometer data,
y
‐component
zc
calibrated magnetometer data,
z
‐component
xnc
uncalibrated magnetometer data,
x
‐component
ync
uncalibrated magnetometer data,
y
‐component
znc
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.
4
Data Acquisition and Analysis
This chapter fundamentally deals with how we'll digitally represent the various aircraft performance characteristics for further analysis. A critical element of this chapter is a detailed discussion of how digital data acquisition (DAQ) systems work. Our discussion of DAQ techniques is motivated by industry and military flight test programs, which rely on complex data acquisition systems. Instrumentation on board the aircraft can involve hundreds or thousands of sensors of various kinds, and an equivalent number of channels to digitize and record this data. Data can be sampled at high rates in order to effectively capture transient phenomena, leading to vast quantities of data which require high bandwidth and storage. Furthermore, these data are often streamed in real time to ground stations via radio telemetry link, such that flight test engineers can monitor the data as the test is being conducted. Live telemetry of data and real‐time analysis adds to the safety and efficiency of the flight test program, enabling the flight test team to avoid hazardous test conditions or to adapt to test events as they develop.
While the typical student will not be able to work with such high‐end systems in the university environment, the basic principles of DAQ are still relevant in low‐cost, small‐scale data acquisition systems. With the continued evolution and miniaturization of digital electronics, these systems are now readily accessible even to students. Simple data acquisition systems in contemporary flight testing use include smartphones (Gregory and Jensen 2012; Gregory and McCrink 2016), LabVIEW‐ or MATLAB‐based digital DAQ systems (Muratore 2012), Arduino microprocessors (Koeberle et al. 2019), commercially available systems, and even the standard avionics onboard the aircraft (see Chapter 3). Thus, students can readily get exposure to the basic principles of DAQ systems, methods, and data analysis employed in flight testing.
This chapter provides an overview of the foundations related to DAQ and processing, such that the capabilities and limitations of DAQ methods can be appreciated. Our discussion of DAQ will begin with defining how signals may be represented as a function of time or frequency. This will directly lead to a discussion of filtering, whereby unwanted frequency content can be attenuated in a signal. Following this, we'll focus on the essential characteristics of DAQ, with an overview of the methods used to digitally represent an analog signal. We'll then conclude with an example of how DAQ techniques are applied to flight testing.
4.1 Temporal and Spectral Analysis
We'll start our discussion of DAQ by considering how signals can be represented in either the time domain or the frequency domain. Both representations of a signal are of use in the flight testing environment. The time domain representation is the most intuitive, where the signal is plotted as a function of time. The frequency domain representation is less intuitive, but no less powerful. This form of presenting a signal shows the relative significance of different frequency components found in the signal, facilitating visual separation of different frequency peaks in the spectrum. We'll examine both approaches of data representation as follows, and establish the link between the two.
Figure 4.1 Sample signal in the time domain.
A time‐domain representation of a signal is our intuitive view of signal waveforms, which is a plot of voltage as a function of time. For example, Figure 4.1 shows a plot of the function
(4.1)
in the time domain, where c0 = 2, c1 = 1, c2 = 0.5, c3 = 0.5, ω1 = 50.3 rad/s (f1 = ω1/2π = 8 Hz), ω2 = 5ω1, and ω3 = 10ω1. A representation of the same signal in the frequency domain, however, will plot the amplitude of each frequency component of the signal versus the corresponding frequency. Inspection of Eq. (4.1) reveals that the signal has three non‐zero frequency components: a dominant, low‐frequency component with amplitude of c1 at a frequency of ω1; and two smaller, higher‐frequency components with amplitudes c2 and c3 at frequencies ω2 and ω3. We can take these three amplitudes (c1, c2, and c3), along with the time‐averaged amplitude of the signal (c0, which has a frequency of ω0 = 0) and plot them versus the corresponding frequencies, resulting in the frequency domain plot shown in Figure 4.2.
Plotting the signal represented in Figure 4.1 in the frequency domain is straightforward in this case since we know the amplitude and frequency of each component from Eq. (4.1). But what if we don't know the frequency content of a signal? How would we generate a frequency domain representation of some arbitrary signal that we have acquired? The answer lies in a computational technique known as the fast Fourier transform (FFT), which is a straightforward and fast algorithm for computing spectral content of digitized signals. This section will provide only a very abbreviated overview of spectral analysis, which is the process of representing a time‐based signal in the frequency domain. The interested reader is encouraged to consult other sources such as Wheeler and Ganji (2003) or Bendat and Piersol (2010) for further details.
The concept behind Fourier analysis is that an arbitrary, periodic signal may be represented by a summation of a constant with an arbitrary number of sine and cosine functions (Powers 1999). The mathematical representation of a Fourier series is given as
(4.2)
Figure 4.2 Sample signal in the frequency domain.
Note that our example function defined earlier (Eq. (4.1)) is of the same form as Eq. (4.2), which made it straightforward for us to pick out the coefficients of the Fourier series by inspection and plot the signal in the frequency domain (Figure 4.2). For an arbitrary periodic waveform, the coefficients in Eq. (4.2) may be determined from
(4.3)
which is the time‐average of the signal over one period of the waveform, T = 1/f = 2π/ω. The other coefficients are given by
(4.4)
Note that Eq. (4.2) is an infinite series, implying that an infinite number of coefficients may be required to fully represent an arbitrary periodic waveform. In practice, the number of coefficients used to represent a signal is truncated to some reasonable number of computationally determined coefficients. If Eqs. (4.2)–(4.4) are evaluated for the sample function given by Eq. (4.1), with T = 2π/ω1 = 0.125 seconds, we could directly compute the integrals and find that the coefficients of the Fourier series are a0 = c0, b1 = c1, b5 = c2, and b10 = c3, with all other coefficients being zero.
Figure 4.3 shows another example of the application of a Fourier series representation to a triangle waveform,
(4.5)
Figure 4.3 Fourier series approximation of a triangle waveform.
with frequency ω = 314.2 rad/s (f = 50 Hz). Note that the function represented in Eq. (4.5) is odd, meaning that f(t) = − f(−t). The implication of this for the Fourier series is that the an coefficients are zero and the function can be represented entirely by the sine terms in Eq. (4.2), making it a Fourier sine series. The coefficients of this sine series are b1 = − 0.8106, b3 = 0.09006, b5 = − 0.03242, b7 = 0.01654, …. Inclusion of each successive term in the sine series improves the fidelity of the Fourier approximation to the original waveform (Figure 4.3), with higher frequency coefficients improving the fit at the peaks of the triangle waveform. This is because the amplitude of the higher order terms decays rapidly with frequency, as shown in Figure 4.4.
Figure 4.4 Fourier components of the sine series approximation to a triangle waveform.
While the definition of the Fourier series is useful for illustrating the representation of signals in the frequency domain, the development thus far is not yet useful for frequency representation of an arbitrary waveform that we might encounter in flight testing. Actual analysis of signals via Fourier techniques is done through the Fourier transform, which relaxes the constraint on periodicity. The steps in frequency (ωn = nω) in Eq. (4.2) are reduced until ω becomes a continuous function of frequency and we have
(4.6)
which is the Fourier transform of a function f(t). Here the exponential function is another way of representing the sine and cosine terms, and
(4.7)
In applying the Fourier transform to a digital representation of a signal (sampled at discrete time intervals) the discrete Fourier transform (DFT) is used. The DFT is represented by
(4.8)
where N is the total number of samples in the data record, Δt is the time interval between samples, Δf is the increment in frequency (equal to the inverse of the period), and there are k frequency components in the transform. (Note that only values up to k = N/2 are unique). In the same way that we defined an inverse Fourier transform, we can also define an inverse DFT,
(4.9)
for recovering the original signal.
The DFT and inverse DFT can be determined numerically, but this process tends to be computationally expensive since the number of computations is on the order of N2 real‐valued multiply‐add operations. Due to this computational expense, the FFT technique has been developed, which requires on the order of Nlog2N computations. (For a record length of N = 216, the FFT requires 212 fewer computations compared to the DFT). Bendat and Piersol (2010) may be consulted for complete details on the derivation and implementation of the FFT algorithm. Returning to our analysis of a triangle waveform (Eq. (4.5)), spectral analysis based on the FFT algorithm results in the power spectrum shown in Figure 4.5. The dominant frequency and higher harmonics are faithfully captured by the FFT algorithm, as is evident when we compare Figure 4.5 with Figure 4.4.
Note that the power spectral density is often represented on a logarithmic scale, which helps the higher‐frequency components of the signal appear more prominently on the plot (this is in contrast to the linear scaling employed in Figures 4.2 and 4.4). Power spectra often have broader frequency peaks along with lower‐amplitude ripple between the peaks, which are characteristic of the DFT and FFT algorithms. This phenomenon, referred to as spectral leakage, results from a non‐integer number of waveforms being present within the data record analyzed by the FFT, with the end effects being the primary culprit. Leakage may be reduced through windowing, where the window is a tapering function (high in the center region, with low values at the ends) multiplied with a subset of the data. Since the window size is a subset of the full data record, the full FFT is then the average of the computed FFTs of the windowed portions of the signal. An amount of overlap may be specified, which determines how much the window is shifted along the length of the record. Welch's modified periodogram method (Welch 1967) is one common approach to implementing a windowing function with overlap (see the
