November 2013

NASA/TM–2013-218056

Dependence of Dynamic Modeling Accuracy

on Sensor Measurements, Mass Properties, and 

Aircraft Geometry

Jared A. Grauer and Eugene A. Morelli

Langley Research Center, Hampton, Virginia
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November 2013

NASA/TM–2013-218056

Dependence of Dynamic Modeling Accuracy

on Sensor Measurements, Mass Properties, and 

Aircraft Geometry

Jared A. Grauer and Eugene A. Morelli

Langley Research Center, Hampton, Virginia
Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802

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National Aeronautics and Space Administration. 
Abstract

The NASA Generic Transport Model (GTM) nonlinear simulation was used to investigate

the effects of errors in sensor measurements, mass properties, and aircraft geometry on the

accuracy of identified parameters in mathematical models describing the flight dynamics

and determined from flight data. Measurements from a typical flight condition and system

identification maneuver were systematically and progressively deteriorated by introducing

noise, resolution errors, and bias errors. The data were then used to estimate nondimensional

stability and control derivatives within a Monte Carlo simulation. Based on these results,

recommendations are provided for maximum allowable errors in sensor measurements, mass

properties, and aircraft geometry to achieve desired levels of dynamic modeling accuracy.

Results using additional flight conditions and parameter estimation methods, as well as a

nonlinear flight simulation of the General Dynamics F-16 aircraft, were compared with these

recommendations.

1
Contents

1 Introduction 5

2 Materials 6

2.1 NASA Generic Transport Model . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Reference Excitation Maneuver . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Methods 7

3.1 Aerodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Equation-Error Parameter Estimation in the Frequency Domain . . . . . . . 8

3.3 Error Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Results 10

4.1 Nominal Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Sensor Resolution Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Mass and Geometry Bias Effects . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.4 Additional Flight Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.5 Additional Parameter Estimation Methods . . . . . . . . . . . . . . . . . . . 14

4.6 F-16 Nonlinear Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Conclusions 16

2
Nomenclature

Roman

A multisine total amplitude [rad]

ax, ay, az body frame translational acceleration [ft/s2]

ak multisine amplitude [rad]

b wingspan [ft]

C nondimensional coefficient

c ̄ mean aerodynamic chord [ft]

e error vector

I inertia [slug · ft2]

J(θ) cost function

j imaginary number, √

−1

K set of available excitation frequencies

M total number of excitation frequencies

m mass [slug]

N number of data samples

p, q, r body frame roll, pitch, yaw rates [rad/s]

q ̄ dynamic pressure [lbf/ft2]

R noise covariance matrix

< real part

S wing area [ft2]

T data record length [s]

t time [s]

V airspeed [ft/s]

X regressor matrix

y model output

z measurement

Greek

α angle of attack [rad]

β sideslip angle [rad]

∆ perturbation value

δ control surface deflection [rad]

θ parameter estimate vector

ρ air density [slug/ft3]

Σ(θ) parameter estimate covariance matrix

σ standard deviation

φ, θ, ψ Euler roll, pitch, yaw angles [rad]

φk multisine phase angle [rad]

ω frequency [rad/s]

Superscripts

 ̇ time derivative

†

complex conjugate transpose

ˆ estimated value

−1

inverse

Subscripts

0 trim value

a aileron

3
e elevator

l

, m, n body frame roll, pitch, yaw moments

r rudder

X, Y , Z body frame longitudinal, side, heave forces

x, y, z body axis components

Acronyms

CAD Computer-Aided Design

GTM Generic Transport Model

NASA National Aeronautics and Space Administration

RPF Relative Peak Factor

SIDPAC System IDentification Programs for AirCraft

4
1 Introduction

A routine goal of system identification, within the context of aircraft flight dynamics, is to

estimate from experimental flight test data a set of nondimensional stability and control

derivatives, and their uncertainties, within a linear system of differential equations. These

results comprise a dynamic model of the aircraft, which then enables applications including

performance and handling qualities analysis, aircraft redesign, flight simulator development,

and control law synthesis. It is important to understand how errors in the flight test and

system identification processes manifest in the parameter estimates to have confidence in

the results and to adequately instrument the hardware. Failure to do so will lead to poorer

system identification results that may require additional time and money to correct.

The consequences of incurring measurement noise, unmodeled dynamics, and information deficiency due to lack of excitation on system identification results are well understood [1–3]. Although not in common practice, methods were developed to identify models

from quantized data [4, 5]. In the 1970’s, studies [6–9] were conducted to determine the

consequence of flight instrumentation errors on the aircraft stability and control derivative

estimates, in dimensional form, using the output-error method in the time domain [1, 10]

with linear simulation models and doublet excitation inputs. These works focused on time

skews; filter lags; sensor position, misalignment, bias, and scale factor errors; and accelerometer and air flow angle correction errors. Because the estimates were in dimensional form,

results were not valid for other flight conditions or for other aircraft. These studies were not

able to take advantage of modern excitation input design and parameter estimation techniques, nor did they consider nondimensionalized stability and control derivatives, nor did

they examine the effect of sensor resolution, mass and inertia errors, and geometry errors.

One sector of current research and development that could benefit from this information

is the design of subscale, unmanned air vehicles. Although envisioned as cheaper alternatives

to manned flight vehicles, these aircraft may become prohibitively expensive due to the cost

of high quality onboard sensors that are also small and light weight. Understanding the

trade offs between sensor quality and modeling accuracy helps designers create better aircraft

system solutions faster and more cheaply. Mass distribution and aircraft geometry properties

are found using intricate computer aided drawing (CAD) models or by experiment, both

of which are time consuming processes that introduce error and must be repeated for new

configurations. Designers would benefit from a set of guidelines and recommendations that

relate the accuracy of these measurements to the accuracy of dynamic models identified

from experimental flight test data.

This report presents such guidelines. Section 2 describes a high-fidelity, nonlinear simulation of a transport style aircraft, as well as a maneuver designed for system identification.

Section 3 presents the model structure for the aerodynamic force and moment coefficients,

and explains the equation-error method for estimating nondimensional stability and control derivatives in the frequency domain using experimental flight test data. In Section 4,

increasing errors are sequentially introduced to each measurement while nondimensional stability and control derivatives are estimated and recorded within a Monte Carlo simulation.

A summarizing table with recommendations to meet 2%, 5%, and 10% error budgets on

the most important stability and control derivatives is presented. This table is the primary

contribution of this paper and will give designers guidelines for selecting instrumentation

and measurement accuracies. These recommendations are compared with additional flight

conditions, parameter estimation methods, and aircraft simulations.

All of the input design, signal processing, and system identification work presented in this

paper was done using software written in MATLAB
Rcalled System IDentification Programs

for AirCraft (SIDPAC) [1], which was developed at NASA Langley Research Center and is

continually being expanded and improved. SIDPAC has been applied successfully to a

wide variety of flight and wind tunnel experiments at NASA Langley Research Center and

5
elsewhere, and is used at more than 80 institutions worldwide.

2 Materials

2.1 NASA Generic Transport Model

This work used the NASA Generic Transport Model (GTM), which is a nonlinear, rigidbody, flight dynamics simulation of a subscale, transport aircraft having mass and geometry

parameters listed in Table 1. The GTM was selected for analysis because it is a high-fidelity,

nonlinear simulation of a subscale aircraft. The GTM represents a conventional transport

aircraft, having relatively well-known flight dynamics and behaviors. Additionally, the GTM

was chosen because parameter estimation results can be compared to the true parameters

determined using numerical linearization.

The GTM simulates the nonlinear, six degree of freedom, rigid body dynamics of the

aircraft. Aerodynamic control surfaces include the elevator, aileron, rudder, spoilers, stabilizers, and flaps. The aerodynamic model used in the GTM is derived from a series of wind

tunnel tests conducted at the NASA Langley Research Center [11]. A 5.5% scaled wind

tunnel test article was used in the 14- by 22-Foot Subsonic Tunnel under static and forced

oscillation testing to generate aerodynamic tables. These tables were augmented with data

from rotary balance tests conducted using a 3.5% scaled wind tunnel test article in the

20-Foot Vertical Spin Tunnel. A polynomial-based aerodynamic model was then extracted

from the data using multivariate orthogonal functions, expanded in the aircraft states and

controls, and is implemented in the simulation [1, 12, 13]. The GTM also includes two turbojet engines, the dynamics for which are modeled as first order lags from the pilot throttle

input to the thrust output. This model is based on multivariate orthogonal functions identified from ground testing, with additional ram drag corrections that vary with altitude and

airspeed.

The GTM simulation software was written in MATLAB
R. The user provides an initial

state vector and a time history of control surface inputs. The simulation then computes time

histories of the aircraft translational and rotational positions, velocities, and accelerations,

as well as power and thrust states. Additional tools are provided to trim the aircraft and

compute linear perturbation models about those trim conditions.

2.2 Reference Excitation Maneuver

Simulation data for a single reference flight maneuver were examined. For this maneuver, the

GTM was trimmed for straight and level flight at a 1200 ft altitude and a 130 ft/s airspeed,

which is a typical flight condition for this aircraft [14]. The simulation code determined trim

values of +4.52 deg angle of attack, +1.40 deg elevator deflection, and 15% throttle for this

condition.

Orthogonal phase-optimized multisine control input perturbations [1,15] were chosen to

excite the GTM for system identification. These multiple-axis inputs are commonly used

for system identification at NASA Langley Research Center and have led to good modeling

results in a variety of flight conditions including hypersonic, high sideslip stall, and post-stall

flight, as well as normal flight conditions [15, 16]. Each of the elevator, aileron, and rudder

control surface deflections had the form

δ(t) = δ0 + ∆δ(t)

= δ0 +

X

k∈K

ak sin 

2πk

T

t + φk



(1)

6
where δ0 is the trim value, ∆δ(t) is the perturbation, ak is the amplitude, T is the excitation

record length, φk is the phase angle, ωk = 2πk/T is the excitation frequency, and K is the

set of available frequencies for that input.

The excitation record length was chosen as T = 35 s, which corresponds to a fundamental

frequency of 0.0286 Hz, to provide sufficient information content for system identification.

Excitation frequencies were selected as integer multiples of this fundamental frequency and

between the range 0.2 Hz and 2.0 Hz, where the rigid body dynamics of interest typically

reside. Because the excitation frequencies are harmonic multiples of the fundamental frequency, the excitation inputs are mutually orthogonal and were excited simultaneously to

shorten experiment durations without correlating data and deteriorating estimation results.

Excitation frequencies were assigned to the inputs in an alternating manner so that each

input has wide-band frequency content. For simplicity, amplitudes were selected to have

uniform power and normalized such that the elevator, aileron, and rudder have perturbation

amplitudes

A = ak

√

M (2)

of 2.0 deg, 0.5 deg, and 1.5 deg, respectively, where M is the number of frequencies used in

the input. These amplitudes are typical of those used in practice [16]; smaller amplitudes

result in lower signal-to-noise ratios and larger amplitudes can excite nonlinear motions.

The phase angles were determined using a simplex search optimization to minimize the

relative peak factors

RPF = max[∆δ(t)] − min[∆δ(t)]

2

√

2 · rms[∆δ(t)]

(3)

of the inputs to keep the aircraft near the trim condition for linear modeling. These excitation inputs are parameterized in Table 2.

The GTM was simulated with these inputs at the reference flight condition and the

resulting time history data is shown in Figure 1. Dynamic responses are small enough to

remain within expected linear regimes and large enough so measurements are above realistic

noise floors. Airspeed, air flow angles, and rotational velocity outputs are all within the

regions of validity for the aerodynamic database [11].

3 Methods

3.1 Aerodynamic Models

Observing the conventional simplifying assumptions for a rigid body flight dynamics model

of a fixed-wing aircraft [1, 17, 18], the nonlinear equations of motion can be rearranged to

compute the nondimensional aerodynamic force and moment coefficients

CY = (may)/( ̄qS)

CZ = (maz)/( ̄qS)

Cl = [Ixxp ̇ − Ixz(  ̇r + pq) + (Izz − Iyy)qr]/( ̄qSb)

Cm = [Iyyq ̇ + (Ixx − Izz)pr + Ixz(p

2 − r2

)]/( ̄qSc ̄)

Cn = [Izzr ̇ − Ixz(  ̇p − qr) + (Iyy − Ixx)pq]/( ̄qSb) (4)

by substituting measured data on the right side of Equation 4 [1]. The longitudinal coefficient CX is not included here because it is more closely associated with the slower phugoid

mode and is often excluded from a control law design. The mass, inertia, wing reference

area, mean aerodynamic chord, and wingspan of the aircraft are typically known before

the flight test. Dynamic pressure, rotational velocities, and translational accelerations are

7
measured by onboard sensors. Angular velocity measurements are numerically smoothed

and differentiated to compute angular accelerations [1].

The aerodynamic force and moment coefficients in Equation 4 were modeled in terms of

the aircraft states and controls with the linear expansions [1, 17, 18]

CY = CYβ∆β + CYp

b∆p

2V

+ CYr

b∆r

2V

+ CYδa ∆δa + CYδr ∆δr

CZ = CZ0 + CZα ∆α + CZq

c ̄∆q

2V

+ CZδe ∆δe

Cl = Clβ∆β + Clp

b∆p

2V

+ Clr

b∆r

2V

+ Clδa ∆δa + Clδr ∆δr

Cm = Cm0 + Cmα ∆α + Cmq

c ̄∆q

2V

+ Cmδe ∆δe

Cn = Cnβ∆β + Cnp

b∆p

2V

+ Cnr

b∆r

2V

+ Cnδa ∆δa + Cnδr ∆δr (5)

which are valid for small deviations about the trim condition, where ∆ indicates a perturbation. This is a commonly used aerodynamic model and application of step-wise regression [1]

confirmed that this model is statistically appropriate for the reference flight maneuver data.

3.2 Equation-Error Parameter Estimation in the Frequency Domain

Given the measurements of the aerodynamic coefficients in Equation 4 and the model structure in Equation 5, it is a parameter estimation problem to determine the unknown nondimensional stability and control derivatives that best match the equations to the data. The

equation-error method in the frequency domain [1] was used to estimate the stability and

control derivatives. This method ignores sensor noise and modeling error at higher frequencies, has produced good results in practice, and provides an analytical solution amenable to

Monte Carlo simulation.

The first step is to transform the data into the frequency domain using the Fourier

transform. A measured signal z(t) has the finite Fourier transform

z(jω) = Z T

0

z(t)e

−jωtdt (6)

which, for relatively fast sampling frequencies, can be approximated by the discrete Fourier

transform

z(jωm) ' ∆t

N

X−1

i=0

z(i∆t)e

−jωmi∆t

(7)

where ∆t is the sampling period and N is the number of data samples. A high-accuracy

chirp-z transform with cubic polynomial interpolation [1, 19] was used to implement the

Fourier transform using frequencies between 0.1 Hz and 2.5 Hz, in 0.025 Hz increments.

Each force and moment coefficient parameter estimation problem in Equation 5 was

arranged into the least-squares framework as

z = y + e

= Xθ + e (8)

where y is the Fourier transform of the model output, e is the Fourier transform of the

modeling error, X contains Fourier transforms of the explanatory variable time histories,

8
and θ are the unknown nondimensional stability and control derivatives. Minimizing the

least-squares cost function

J(θ) = 1

2

(z − Xθ)

†

(z − Xθ) (9)

where † denotes the complex conjugate transpose, results in the parameter estimates and

associated uncertainties

θˆ =



<{X†X}

−1

<{X†z}

Σˆ(θˆ) = ˆσ

2



<{X†X}

−1

(10)

where the equation-error variance ˆσ

2 was estimated from the model residuals [1] and <

denotes the real part of a complex number. The bias terms CZ0

and Cm0

cannot be estimated

with frequency domain techniques because the data is detrended prior to analysis, but these

terms are not used in a linear dynamics model.

3.3 Error Models

This section describes the mathematical models for the errors used to corrupt the measurements. The first type of error examined is due to quantization, which occurs when sensor

measurements are converted into digital numbers. Quantization is an important issue in

system identification because it changes the amplitude and frequency content of the data.

This source of error remains a problem because although data systems can sample over a

large range, system identification maneuvers are often designed to elicit small-amplitude

responses that can be near the resolution floor of the avionics hardware. Errors due to

quantization affect the computation of the aerodynamic coefficients in Equation 4 and the

nondimensional stability and control derivatives in Equation 5. For a given measurement

z(t), the quantized measurement is

quant{z(t)} = ∆z · round 

z(t)

∆z



(11)

where ∆z is the resolution of the measurement and round is a function that rounds the

argument to the nearest integer. Figure 2 shows for example the progressive quantization

of the angle of attack measurement from the reference excitation maneuver. The first plot

is the original time history with added measurement noise. The next three plots show this

measurement as the resolution ∆α is degraded to 0.6 deg, 1.2 deg, and 1.8 deg.

The second source of error considered is bias error in the aircraft mass distribution and

geometry parameters. The mass distribution of the aircraft can be measured experimentally

using balancing and swinging techniques, or can be computed using CAD software. In either

case, errors occur in determining the aircraft mass and the inertia tensor. Similarly, errors

are incurred in the description of the aircraft geometry, namely the mean aerodynamic

chord, the wingspan, and the wing reference area. These parameters are important because

they affect the computations in Equation 4 and 5. For example, the pitch moment of inertia

is modeled as Iyy + ∆Iyy, where ∆Iyyis a constant, uniformly distributed bias.

Several error models were not considered in this report. While realistic noise sequences

were applied to the data, the amplitude and statistical characteristics of the noise sequences

were not varied because analytical and experimental results for describing the accuracy of

estimated parameters using noisy data already exist [1–3]. Other studies [6,8,9] investigated

the effects of bias and scale factor errors on sensor calibrations, as well as alignment, installation, and correction errors. In the frequency domain analysis used here, bias errors are all

removed during detrending, before the Fourier transforms are applied, and are not a factor

in the linear dynamic modeling. Scale factor errors manifest as multiplicative errors which

9
are directly proportional to the estimated stability and control derivatives. Finally, a study

using output-error parameter estimation was already conducted to investigate the effects

due to time skews between sensor measurements and signal-conditioning filter lags [7].

4 Results

This section presents the results of dynamic modeling accuracy dependence on sensors measurements, mass properties, and aircraft geometry. A baseline case is first presented, where

nondimensional stability and control derivatives were estimated without corrupting the measurements, except for adding Gaussian white noise. Afterwards, errors were systematically

introduced into the measurements to determine thresholds for meeting 2%, 5%, and 10%

accuracies on the estimated parameters. Last, the analysis was repeated using different trim

conditions, parameter estimation methods, and aircraft simulations to investigate sensitivities to these changes.

Monte Carlo simulations were used to produce the results. During investigation of the

measurements of the aircraft mass properties or geometry, these values were first given a

known random bias, distributed uniformly on a domain described by a percentage of the

nominal value. Then measurements of the system outputs, for example the angle of attack

and sideslip angle, were corrupted with a random Gaussian white noise sequence that implements a signal-to-noise ratio of 20:1, based on the root mean square of the variation in the

data. The elevator, aileron, and rudder deflections were not corrupted with noise because

these measurements are typically of high quality. During investigation of sensor measurements, the appropriate signals were then quantized. One thousand runs were performed for

each level of accuracy and for each measurement corrupted. This number of runs was found

to be beyond the amount needed for statistical convergence.

The estimated nondimensional stability and control derivatives diverged from the nominal estimates and exhibited increasing variation as the measurements were progressively

corrupted. The damping derivatives, e.g. Cnr, and off-axis derivatives, e.g. Cnδa, are less

important in the dynamic model and are more difficult to estimate using system identification because the dynamic responses are smaller. Therefore, only the more important

on-axis derivatives, e.g. Cnδr, are reported. Stability and control derivative estimates are

plotted versus the accuracy level, as shown, for example, in Figure 4. Each point represents

the mean value of the Monte Carlo simulations, using one thousand different measurement

corruptions and noise sequences. The error bounds on each point represent the two standard

deviation spread of the Monte Carlo simulations. The dashed lines represent the 10% variation of the finite difference results. Perturbations for computing numerical finite difference

values were selected as the maximum deviations from the trim values in the flight data.

Nondimensional stability and control derivatives computed using finite differences are listed

in Table 3. Once any part of the two standard deviation interval crossed the 10% variation

line for any of the reported on-axis stability and control derivatives, that level of accuracy

was reported in Table 4 as the minimum needed to meet 10% error. For clarity, the 2% and

5% error bound levels are not shown in the plots, but those results are also listed in Table

4. This table, describing minimum accuracy levels for several measurements, is the main

contribution of this work.

4.1 Nominal Estimation Results

Equation-error parameter estimation was used in the frequency domain with the reference

flight data to estimate the nondimensional stability and control derivatives in the aerodynamic model. Figure 3(a) shows the model fits to the force and moment coefficients in the

frequency domain, and Figure 3(b) shows the residuals, which are plotted at one tenth the

10
amplitude scale. The model fits had coefficients of determination above 0.99, the residuals

had standard deviations two orders of magnitude less than the data, and estimated error

bounds were all within 3.7% of the estimated model parameters. These metrics indicate

that the model is appropriate for this set of data.

Nominal estimates of the nondimensional stability and control derivatives are provided

in Table 3. The equation-error results were within two standard deviations of the finite

difference results and thus were in statistical agreement. Differences between the results are

mainly due to measurement noise and linearization effects. Table 3 also shows stability and

control estimates using other methods of parameter estimation. Since all of these results

are in statistical agreement, these results are consistent and there is sufficient information

content in the data to identify accurate dynamics models.

4.2 Sensor Resolution Effects

This section discusses the effects of sensor quantization on the accuracy of dynamic models.

Figures 4 through 11 illustrate the results and Table 4 contains the values at which the 2%,

5%, and 10% error criteria are violated.

When frequency domain equation-error parameter estimation is applied, resolution errors

manifest as increased biases in the parameter estimates, shown in the figures as diverging

oscillations. Although these results were obtained using Monte Carlo simulation with unique

noise sequences, there is very little spread in the results, as indicated by the small error

bounds, suggesting low sensitivity to varying random noise sequences at the noise level used

here.

Overall, the longitudinal dynamics were more robust to quantization errors than the

lateral/directional dynamics because the lateral/direction dynamics are more tightly coupled

and contain more parameters to estimate. When one parameter begins to diverge, the other

parameters become biased and also diverge. The damping derivatives depart first, followed

by the control derivatives and then the static derivatives. Damping derivatives are typically

the most difficult to estimate in aircraft system identification, whereas static derivatives

are strong parameters and there is good excitation available for estimating the control

derivatives.

The first measurements investigated were the angle of attack and sideslip angle, which

are made with air flow vanes. These are important measurements because the aerodynamic

forces and moments depend strongly on them. Corrections are often made for bias and scale

factor calibrations, as well as offsets from the center of gravity. These measurements do not

affect the aerodynamic coefficient calculations in Equation 4, but serve as explanatory variables in Equation 5. As the quantization level increases, making measurements of the angle

of attack and sideslip angles progressively more coarse, the average values of the estimated

stability and control derivatives from the Monte Carlo simulations diverge from the original

values. The remainder of the linear nondimensional stability and control derivatives are

shown in Figure 5 for completeness. As these are weaker parameters, they diverge sooner

than the derivatives shown in Figure 4. Some of these parameters, for example Cnδa, violate

the 10% error criterion even before quantization.

Gyroscopes measure the body-fixed rotational velocities. They are subject to bias errors

and drifts, alignment errors, and effects due to structural flexibility of the aircraft. Gyroscope measurements and their derivatives are used in both the aerodynamic coefficients and

the aerodynamic model. Figure 6 shows that the static terms CYβ

and CZα are relatively

robust to gyroscope errors. In this case Clδashows a particular sensitivity to measurement

noise, as indicated by the larger spread in the Monte Carlo results.

Accelerometers measure the translational accelerations of the body and are subject to

bias errors, position errors due to sensor offset from the center of mass, and alignment errors.

Accelerometers only affect the computation of the aerodynamic force coefficients CY and

11
CZ in Equation 4. Results are shown in Figure 7. The derivative CYδr

limits the resolution

because the side acceleration is much smaller than the heave acceleration, as seen in the

flight data in Figure 1.

Angle of attack and sideslip angle measurements can also be reconstructed when flying

about trimmed flight conditions [20], such as the reference maneuver studied here. The

differential equations

β ̇ ' αp − r + (sin φ cos θ + ay)/V

α ̇ ' q − βp + (cos φ cos θ + az)/V (12)

may be integrated from initial conditions to reconstitute air flow angle measurements. While

bias and drift errors accumulate during numerical integration of noisy data, these are removed prior to applying the Fourier transformation and do not affect the parameter estimates determined in the frequency domain. Figures 8 and 9 show nondimensional stability

and control derivative estimates using reconstructed air flow angles and quantized gyroscope

and accelerometer measurements, respectively. Corruption of the airspeed measurement in

Equation 12 did not significantly deteriorate the estimates. For the stability and control

derivatives under consideration, applying airflow angle reconstruction with degraded sensor

resolutions quickly violated the error bound criterion due to additional biases and increases

in the spread of the Monte Carlo simulation results. Here, not only are the rotational velocity and linear acceleration measurements degraded, but also the reconstructed air flow angle

measurements. The lateral/directional derivatives were more susceptible than the longitudinal derivatives, and the air flow angle derivatives were more susceptible than the control

derivatives. While stability and control derivative estimates were accurate, reconstructing

air flow angle data relies on less information and is more susceptible to errors in this data,

as previously noted [20].

Airspeed measurements are computed from pressure measurements, which are usually

corrupted by bias and scale factor errors. Dynamic pressure measurements are used to

nondimensionalize the aerodynamic coefficients. Figure 10 shows that after about 5 lbf/ft2

quantization error in dynamic pressure, the parameter estimates diverge quickly. The limiting factor, CZδe

, is the most sensitive because there is high excitation in the heave acceleration, and errors in the dynamic pressure magnify errors in the parameter estimation.

Control surface deflections are measured with potentiometers, and these are typically

low in noise and correction errors. These measurements are of paramount importance in

system identification, especially when using methods such as output-error that integrate the

equations of motion using the input time histories. In the equation-error approach, these

measurements only appear as explanatory variables in the aerodynamic model and the

associated estimated model parameters have a reduced sensitivity to error. Figure 11 shows

that the control derivative estimates diverge first as the quantization error in the control

surface position measurement increases. After quantization at 1.4 deg, the contributions

due to the aileron can no longer be deciphered.

It is important to note here that the error decreases with the square root of the number

of data points. If time and money allowed for repeated testing at the same flight condition,

additional data could help mitigate the effects of poor measurement resolution.

4.3 Mass and Geometry Bias Effects

This section discusses the effects on the dynamic model parameters due to errors in measuring the mass and geometry properties of the aircraft. Figures 12 through 14 illustrate

the results and Table 4 contains the values at which the 2%, 5%, and 10% error criteria are

violated.

Figures 12 through 14 show that mass and geometry errors do not change the mean

value of the nondimensional stability and control derivative estimates and instead increase

12
the spread of the results. The longitudinal and lateral/directional derivatives are affected

similarly, unlike the sensor quantization results. The static, damping, and control derivative

uncertainties all grow at approximately the same rate. Due to some biases on the damping

derivatives, these will surpass the error criteria quickly, and so the same set of on-axis static

and control derivatives illustrated in Figure 4 were used to create Table 4.

Figure 12 shows the results for increasing errors in the aircraft mass and inertias, where

the errors were modeled as uniformly distributed, random biases. In practice, these values

are determined by CAD models and/or experimental techniques, and errors directly alter

the aerodynamic coefficients. Errors would also affect center of mass corrections performed

on many of the sensors, but this is not modeled here because the effects would be very small.

There is essentially a one-to-one correspondence between the mass property error and the

parameter estimation error; for example, 10% error in the mass quantities results in the two

standard deviation spread of CZδe surpassing the 10% error criterion. However, the values

presented in Table 4 are less than this because parameter estimates are biased from the

finite difference values.

The product of inertia Ixz is difficult to obtain accurately from experiments, but not

a problem when using CAD modeling. Figure 13 was generated to address whether this

quantity is needed for accurate dynamic modeling. The product of inertia Ixz only affects

the Cl and Cn moment coefficients in Equation 4, and while it did not increase the rate of

growth of the error bounds of the estimated parameters, it led to larger differences between

the Monte Carlo and finite difference results, which in turn led to lower error requirements

for meeting the same error criteria.

Figure 14 shows the results when corrupting the aircraft geometry measurements, which

are either measured directly or computed using CAD programs. These values nondimensionalize the aerodynamic coefficients and appear in the nondimensional rate terms in the

aerodynamic model. Errors impact the moment coefficients more than the force coefficients

because the wing reference area is multiplied by either the mean aerodynamic chord or the

wing span, thereby magnifying the error.

4.4 Additional Flight Conditions

The previous analysis was performed on the GTM nonlinear flight simulation for one single

maneuver. To investigate how the measurement recommendations might change with the

flight condition, the analysis was repeated for trim angles of attack between 0 deg and 10

deg, as described in Table 5. The same input sequence described in Table 2 was applied

to the GTM in each trim condition. As the trim angle of attack increased and the trim

airspeed decreased, the perturbations in the angle of attack increased while the rotational

velocities became smaller. All flight data remained within the regime of validity for the

GTM simulation.

Figure 15 shows estimation results for the pitching moment derivatives as the gyroscope

resolution is progressively degraded, for several trim conditions. The 10% error criteria is

based on the finite difference values listed in Table 5. At 0 deg angle of attack, these values

are surpassed sooner because additional biases exist in the estimated parameters, perhaps

due to the larger excursions from trim experienced. As the trim angle of attack increases, the

manner in which the estimated stability and control derivatives diverge remains the same,

and the values at which the error criteria are violated remain consistent. Therefore, the

recommendations provided in Table 4 are independent of the flight condition, for straight

and level flight below stall. If the model structure in Equation 5 changes, for example

with unusual aircraft configurations or trim conditions, then these values would have to be

determined again.

13
4.5 Additional Parameter Estimation Methods

The previous results employed the equation-error method in the frequency domain. Another

common method for parameter estimation is to apply equation-error in the time domain.

The same procedure is used, except that the Fourier transform in Equation 7 is not applied.

While this process still retains the analytical solution, there are several disadvantages to

the time domain analysis. For instance, more data points are used, resulting in longer

computation times. There is no automatic filtering of frequency content away from the

rigid body dynamics of interest, so higher frequency content such as noise and unmodeled

dynamics must first be filtered. Time-domain residuals are typically colored, so that the

estimated parameter error bounds must be corrected [1].

Results for degrading air flow angle measurements for the GTM reference maneuver are

shown in Figure 16 using equation-error method in the time domain. The on-axis parameters

diverged much faster than using the equation-error method in the frequency domain. The

spread of some parameters including Clβand Clδaviolate the 10% error criteria even with

no quantization, due to the added noise on the measurements.

Another popular method of parameter estimation is the output-error method in the

time domain [1, 10]. In this approach, a nonlinear optimization is used to iteratively find

the stability and control derivatives that best match the model outputs to the experimental

data. Equations 14 through 17 describe the equations of motion for the longitudinal and

lateral/directional variables, as well as the model outputs. Again, the phugoid mode was not

modeled. The output-error method integrates these equations from initial conditions using

the time history of the inputs to obtain the outputs. The stability and control derivatives

are then optimized so that the model outputs best match the measured outputs according

to the cost function

J(θ) = 1

2

(z − y)

T Rˆ −1

(z − y) (13)

where R is the noise covariance matrix.

The results of using output-error estimation in the time domain for the GTM reference maneuver and air flow angle quantization are shown in Figure 17. The output-error

method worked very well in the case of the CZ and Cm coefficients. However as the quantization increased up to 1 deg, the estimates of the lateral derivatives quickly degraded.

The lateral/directional dynamics are more coupled and there are more parameters to estimate, which degraded sooner than using the frequency domain equation-error method. For

the case of quantized input measurements, the output-error method performed particularly

poorly because of the numerical integration involved, consistent with previous research.

4.6 F-16 Nonlinear Simulation

A nonlinear simulation of the General Dynamics F-16 fighter aircraft was also used to check

if results change with the aircraft size. The F-16 is a piloted, full-scale, fighter aircraft with

properties listed in Table 1. The nonlinear simulation is a MATLAB
R implementation [1]

of the FORTRAN simulation presented by Stevens and Lewis [18], which utilizes modified

aerodynamic tables from wind tunnel tests and engine dynamics identified from ground

tests [21].

The F-16 simulation was trimmed at 10,000 ft altitude, with an airspeed of 450 ft/s and

a 5 deg angle of attack. Perturbation inputs described in Table 2 were super-imposed on

trim settings for the nonlinear F-16 simulation.

Parameter estimation results using equation-error in the frequency domain were similar

to those found using the GTM. A representative case, showing parameter estimates with

varying mass and inertia accuracy, is shown in Figure 18, exhibiting similar trends for

varying flight conditions. The use of nondimensional stability and control derivatives and

14
Longitudinal equations:

 ∆  ̇α∆  ̇q  = "  ̄q0S ̄c mV0 CZα + g sin α0 V0 − T0 cos α0 mV0 1 +  ̄q0S ̄c 2mV 20 cos α0CZq  ̄q0S ̄c Iyy Cmα  ̄q0S ̄c2 2V0Iyy Cmq #  ∆α∆q  + "  ̄q0S mV0 CZδe  ̄q0S ̄c Iyy Cmδe # ∆δe (14)



∆α

∆q

∆az

 =  1 0 0 1  ̄qSmg CZα  ̄qS ̄c 2V0mg CZq   ∆α∆q  +  00  ̄qSmg CZδe  ∆δe (15)

Lateral equations:

1 0 0 00 Ixx −Ixz 0

0 −Ixz Izz 0

0 0 0 1





∆

 ̇

β

∆  ̇p

∆  ̇r

∆

 ̇

φ

 =

  ̄qS mV0 CYβ  ̄qSb 2mV 20 CYp + sin α0  ̄qSb 2mV 20 CYr − cos α0 g cos θ0 V0  ̄qSbClβ  ̄qSb2 2V0 Clp  ̄qSb2 2V0 Clr 0  ̄qSbCnβ  ̄qSb2 2V0 Cnp  ̄qSb2 2V0 Cnr 0 0 1 tan θ0 0   ∆β ∆p ∆r ∆φ +  ̄qS mV0 CYδa  ̄qS mV0 CYδr  ̄qSbClδa  ̄qSbClδr  ̄qSbCnδa  ̄qSbCnδr 0 0   ∆δa ∆δr 

(16)

 ∆β ∆p ∆r ∆φ ∆ay  =  1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1  ̄qSmg CYβ  ̄qSb 2V0mg CYp  ̄qSb 2V0mg CYr 0   ∆β ∆p ∆r ∆φ  +  0 0 0 0 0 0 0 0  ̄qSmg CYδa  ̄qSmg CYδr   ∆δa ∆δr  (17)

15
percent errors allow recommendations in Table 4 to be applicable to other aircraft. This

is expected because the nondimensional stability and control derivatives remove the known

dependence on aircraft mass and geometry properties.

5 Conclusions

This report investigated the effects of errors in sensor measurements, mass properties, and

aircraft geometry on the accuracy of dynamic model parameters estimated from experimental flight test data. The NASA GTM nonlinear simulation was excited using orthogonal

phase-optimized multisine perturbations on the elevator, aileron, and rudder inputs while

in trimmed flight. Each case was run one thousand times using unique noise sequences, and

the equation-error method was applied in the frequency domain to estimate longitudinal

and lateral/directional nondimensional stability and control derivatives.

The findings and contributions of this report are:

1. A list of recommended measurement resolutions needed to meet 2%, 5%, and 10%

errors budgets on estimated nondimensional stability and control derivatives using

equation-error in the frequency domain are provided in Table 4.

2. Estimates of stability and control derivatives can be obtained by reconstructing angle of attack and sideslip angle measurements from gyroscopes and accelerometers;

however, these sensors require better resolution in this case, especially the gyroscopes.

Nonetheless, this method appears to be a viable alternative to using air flow angle

vanes for system identification, as these sensors disturb the flow, add cost and complexity to the aircraft, and are difficult to calibrate.

3. The product of inertia Ixz was found to have a small impact on stability and control

derivative estimates for typical fuselage-heavy aircraft. Here, this quantity can be

safely approximated or neglected to save time and money if aircraft mass distributions

are being determined experimentally, assuming small perturbation flight tests starting

from wings-level flight conditions at relatively low angles of attack.

4. Repeated analysis showed that the results listed in Table 4 were not sensitive to

variations in trim conditions below stall.

5. Equation-error parameter estimation in the frequency domain produces better results

than in the time domain, due to noise sensitivities and unmodeled dynamics. Except

for quantized control surface deflections, the output-error method produced excellent

results up to a certain threshold, when large biases and standard deviations were

incurred.

6. Results in Table 4 were found to scale with vehicle size, per analysis using an F-16

simulation.

16
References

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Education Series, AIAA, 2006.

2. Ljung, L.: System Identification: Theory for the User . Information and System Sciences

Series, Prentice Hall, Upper Saddle River, NJ, 2nd ed., 1999.

3. Morelli, E.; and Smith, M.: Real-Time Dynamic Modeling: Data Information Requirements and Flight Test Results. Journal of Aircraft, vol. 46, no. 6, Nov.–Dec. 2009,

pp. 1894–1905.

4. Wang, L.; Yin, G.; Zhang, J.; and Zhao, Y.: System Identification with Quantized

Observations. Springer, 2010.

5. Konishi, K.; and Kato, H.: A System Identification Method With Roughly Quantized

Data Using Semidefinite Programming. Industrial Electronics Conference, IEEE, 2009,

pp. 1426–1431.

6. Sorenson, J.: Analysis of Instrument Error Effects on the Identification Accuracy of

Aircraft Parameters. TN-D-7830, NASA, Washington D.C., May 1972.

7. Steers, S.; and Illif, K.: Effects of Time-Shifted Data on Flight-Determined Stability

and Control Derivatives. TN–D–7830, NASA, Washington, D.C., March 1975.

8. Hodge, W.; and Bryant, W.: Monte Carlo Analysis of Innacuracies in Estimated Aircraft

Parameters Caused by Unmodeled Flight Instrumentation Errors. TN-D-7712, NASA,

Washington D.C., February 1975.

9. Sorenson, J.; Mohr, R.; and Cline, T.: Instrumentation Requirements for Aircraft Parameter Identification with Application to the Helicopter. CR-132675, NASA, Washington D.C., June 1975.

10. Maine, R.; and Iliff, K.: Application of Parameter Estimation to Aircraft Stability and

Control: The Output-Error Approach. Reference Publication 1168, NASA, June 1986.

11. Murch, A.; and Foster, J.: Recent NASA Research on Aerodynamic Modeling of PostStall and Spin Dynamics of Large Transport Aircraft. No. 2007–463 in 45th AIAA

Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, January 2007.

12. Morelli, E.: Global Nonlinear Aerodynamic Modeling using Multivariate Orthogonal

Functions. Journal of Aircraft, vol. 32, no. 2, March–April 1995, pp. 270–277.

13. Morelli, E.; and DeLoach, R.: Wind Tunnel Database Development using Modern Experiment Design and Multivariate Orthgonal Functions. No. 2003–653 in 41st AIAA

Aerospace Sciences Meeting and Exhibit, AIAA, Reno, NV, January 2003.

14. Jordan, T.; Foster, J.; Bailey, R.; and Belcastro, C.: AirSTAR: a UAV Platform for

Flight Dynamics Control System Testing. No. 2006–3307 in Aerodynamic Measurement

Technology and Ground Testing Conference, AIAA, San Francisco, CA, June 2006.

15. Morelli, E.: Flight-Test Experiment Design for Characterizing Stability and Control

of Hypersonic Vehicles. Journal of Guidance, Control, and Dynamics, vol. 32, no. 3,

May–June 2009, pp. 949–959.

16. Morelli, E.: Flight Test Maneuvers for Efficient Aerodynamic Modeling. No. 2011–6672

in AIAA Atmospheric Flight Mechanics Conference, AIAA, Portland, OR, August 2011.

17
17. McRuer, D.; Ashkenas, I.; and Graham, D.: Aircraft Dynamics and Automatic Control.

Princeton Universtiy Press, 1973.

18. Stevens, B.; and Lewis, F.: Aircraft Control and Simulation. Wiley, 1992.

19. Morelli, E.: High Accuracy Evaluation of the Finite Fourier Transform Using Sampled

Data. TM-110340, NASA, Hampton, VA, June 1997.

20. Morelli, E.: Real-Time Aerodynamic Parameter Estimation Without Air Flow Angle

Measurements. Journal of Aircraft, vol. 49, no. 4, July-August 2012, pp. 1064–1074.

21. Nguyen, L.; Ogburn, M.; Gilbert, W.; Kibler, K.; Brown, P.; and Deal, P.: Simulation

Study of Stall/Post-Stall Characteristics of a Fighter Airplane with Relaxed Longitudinal Static Stability. Technical Report TP 1538, NASA, 1979.

18
Tables

Table 1. Mass and geometry values used in aircraft simulations

Parameter Symbol GTM F-16 Units

center of mass position xcg 0.25 0.25 % ̄c

mean aerodynamic chord  ̄c 0.9153 11.320 ft

wing span b 6.8488 30.000 ft

wing reference area S 5.9018 300.00 ft2

mass m 1.5416 637.16 slug

roll inertia Ixx 1.3270 9496.0 slug·ft2

pitch inertia Iyy 4.2540 55814 slug·ft2

yaw inertia Izz 5.4540 63100. slug·ft2

roll/yaw coupling inertia Ixz 0.1200 982.00 slug·ft2

19
Table 2. Orthogonal phase-optimized multisine input description (record length T = 35 s)

Elevator Aileron Rudder

A = 2.0 deg A = 0.5 deg A = 1.5 deg

ak = 0.6667 deg ak = 0.1091 deg ak = 0.3273 deg

M = 22 M = 21 M = 21

RPF = 1.2445 RPF = 1.2136 RPF = 1.0658

Index Frequency Phase Index Frequency Phase Index Frequency Phase

k k/T [Hz] φk [rad] k k/T [Hz] φk [rad] k k/T [Hz] φk [rad]

7 0.2000 3.6221 8 0.2286 3.9280 9 0.2571 5.8733

10 0.2857 4.1272 11 0.3143 6.0649 12 0.3429 4.9406

13 0.3714 3.2037 14 0.4000 0.8306 15 0.4286 3.1968

16 0.4571 2.6584 17 0.4857 1.5477 18 0.5143 0.0323

19 0.5429 0.0811 20 0.5714 2.2952 21 0.6000 3.3582

22 0.6286 2.5869 23 0.6571 1.4435 24 0.6857 4.3454

25 0.7143 6.0820 26 0.7429 6.2755 27 0.7714 3.9451

28 0.8000 0.0769 29 0.8286 3.1900 30 0.8571 5.7802

31 0.8857 1.6735 32 0.9143 0.0810 33 0.9429 4.4729

34 0.9714 1.7690 35 1.0000 2.0319 36 1.0286 3.3028

37 1.0571 0.6256 38 1.0857 3.0491 39 1.1143 0.0384

40 1.1429 5.0289 41 1.1714 3.8835 42 1.2000 3.6866

43 1.2286 2.5389 44 1.2571 2.3799 45 1.2857 4.8036

46 1.3143 5.3888 47 1.3429 1.6406 48 1.3714 0.3270

49 1.4000 0.8353 50 1.4286 5.3963 51 1.4571 0.8943

52 1.4857 0.6303 53 1.5143 2.3521 54 1.5429 1.1219

55 1.5714 2.2350 56 1.6000 5.2871 57 1.6286 5.7885

58 1.6571 2.3720 59 1.6857 1.8165 60 1.7143 3.7299

61 1.7429 0.9076 62 1.7714 1.7858 63 1.8000 5.4749

64 1.8286 5.1943 65 1.8571 0.6440 66 1.8857 2.8322

67 1.9143 1.2656 68 1.9429 5.5245 69 1.9714 3.7967

70 2.0000 3.4607

20
Table 3. Nondimensional stability and control derivative estimates for the GTM (h0 = 1200

ft, V0 = 130 ft/s, α0 = 4.52 deg)

Numerical Central Frequency Domain Time Domain Time Domain

Parameter Finite Difference Equation-Error Equation-Error Output-Error

θ

ˆθ ± σˆ(ˆθ)ˆθ ± σˆ(ˆθ)ˆθ ± σˆ(ˆθ)

CYβ −1.0125 −1.0065 ± 0.0037 −1.0066 ± 0.0022 −1.0084 ± 0.0124

CYp +0.0543 +0.0735 ± 0.0158 +0.0727 ± 0.0095 +0.0903 ± 0.0553

CYr +0.8574 +0.8877 ± 0.0238 +0.8863 ± 0.0139 +0.9548 ± 0.0761

CYδa −0.0177 −0.0155 ± 0.0032 −0.0156 ± 0.0020 −0.0315 ± 0.0136

CYδr +0.3387 +0.3390 ± 0.0012 +0.3390 ± 0.0006 +0.3446 ± 0.0027

CZα −4.8370 −4.8345 ± 0.0077 −4.8344 ± 0.0040 −4.8932 ± 0.0179

CZq −27.102 −27.609 ± 0.5040 −27.590 ± 0.2823 −28.735 ± 0.9357

CZδe −0.4807 −0.4801 ± 0.0093 −0.4799 ± 0.0052 −0.4540 ± 0.0187

Clβ −0.1432 −0.1393 ± 0.0029 −0.1396 ± 0.0017 −0.1393 ± 0.0021

Clp −0.3542 −0.3333 ± 0.0123 −0.3342 ± 0.0073 −0.3322 ± 0.0096

Clr +0.1331 +0.1535 ± 0.0186 +0.1518 ± 0.0110 +0.1645 ± 0.0140

Clδa −0.0760 −0.0735 ± 0.0025 −0.0736 ± 0.0015 −0.0722 ± 0.0024

Clδr +0.0290 +0.0293 ± 0.0009 +0.0293 ± 0.0005 +0.0291 ± 0.0006

Cmα −1.6349 −1.6134 ± 0.0068 −1.6129 ± 0.0051 −1.6241 ± 0.0053

Cmq −41.215 −41.292 ± 0.4441 −41.301 ± 0.3319 −42.008 ± 0.3408

Cmδe −1.7744 −1.7924 ± 0.0082 −1.7925 ± 0.0061 −1.8238 ± 0.0061

Cnβ +0.2165 +0.2147 ± 0.0009 +0.2150 ± 0.0006 +0.2180 ± 0.0023

Cnp −0.0408 −0.0461 ± 0.0039 −0.0449 ± 0.0024 −0.0366 ± 0.0106

Cnr −0.3840 −0.3892 ± 0.0059 −0.3875 ± 0.0037 −0.3796 ± 0.0159

Cnδa −0.0025 −0.0032 ± 0.0008 −0.0031 ± 0.0005 +0.0027 ± 0.0025

Cnδr −0.1691 −0.1692 ± 0.0003 −0.1691 ± 0.0002 −0.1704 ± 0.0005

Table 4. Recommended measurement resolutions for achieving specified error budgets

Measurement Symbol 2% 5% 10% Units

Air Flow Angles α, β 0.1014 0.2024 0.4048 deg

Dynamic Pressure  ̄q 0.0089 0.1446 2.2406 lbf/ft2

Gyroscopes p, q, r 1.8707 4.2064 5.5439 deg/s

Accelerometers ax, ay, az 0.0127 0.0262 0.0524 g

Potentiometers δe, δa, δr 0.0389 0.2457 0.4526 deg

Mass Distribution m, Ixx, Iyy, Izz, Ixz 0.0057 0.1858 4.1271 %

Mass Distribution m, Ixx, Iyy, Izz, Ixz = 0 0.1798 1.0323 1.4596 %

Aircraft Geometry  ̄c, b, S 0.2684 1.4223 3.0958 %

Table 5. Additional GTM trim conditions (h0 = 1200 ft) and finite difference values

α0 [deg] V0 [ft/s] δt0δe0[deg] Cmα Cmq Cmδe

0 338.33 0.72 +5.19 −1.6437 −44.889 −1.8122

2 177.10 0.24 +3.56 −1.5378 −43.258 −1.7993

4 136.42 0.15 +1.84 −1.6211 −41.628 −1.7750

6 115.51 0.17 +0.18 −1.4117 −40.027 −1.8237

8 102.39 0.15 −1.00 −0.9992 −38.425 −1.8550

10 092.97 0.14 −1.79 −0.8814 −31.095 −1.8230

21
Figures

22
[deg] eδ

[deg] aδ

[deg] rδ

[deg] φ

[deg] θ

[deg] ψ

[ft/s] V

[deg] α

[deg] β

[deg/s] p

[deg/s] q

[deg/s] r

[g] xa

t [s]

[g] ya

t [s]

[g] z a

t [s]

-2420

-2420

-2420

-100

10

-100

10

-10-200

135130125

50

10

-550

-500

50

-500

50

-500

50

-0.50

0.5

0 10 20 30 40

-0.50

0.5

0 10 20 30 40

-1.5

-1

-0.5

0 10 20 30 40

Figure 1. GTM reference maneuver simulation data

23
α [deg]

∆α = 0.0

α [deg]

∆α = 0.6

α [deg]

∆α = 1.2

α [deg]

∆α = 1.8

t [s]

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

0

2

4

6

8

0 5 10 15 20 25 30 35 40

Figure 2. Noisy angle of attack measurement with progressive quantization

24
CY CZ Cl C

m

Cn

f [Hz]

data model

0.00

0.10

0.20

0.00

0.40

0.80

0.00

0.01

0.02

0.00

0.20

0.40

0.00

0.03

0.06

0 0.5 1 1.5 2 2.5

(a) computed and modeled aerodynamic force and moment

coefficients

CY − ˆCY CZ − ˆCZ Cl − ˆCl C

m

− ˆC

m

Cn − ˆCn

f [Hz]

0.000

0.010

0.020

0.000

0.040

0.080

0.000

0.001

0.002

0.000

0.020

0.040

0.000

0.002

0.004

0 0.5 1 1.5 2 2.5

(b) residuals

Figure 3. Frequency domain data modeling for the GTM reference excitation maneuver

25
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg]

δr

n C

quantization level [deg]

-1.5

-1

-0.5

0

0.3

0.35

0.4

0.45

-4-5-6 -1

-0.5

0

-0.1-0.2

0

-0.1

-0.05

0

-2

-1.5

-1

-2

-1.2-1.6

0

0.30.20.1

0 0.5 1 1.5 2

-0.2

-0.14-0.16-0.18

0 0.5 1 1.5 2

Figure 4. Variation in GTM stability and control derivatives with air flow angle resolution

26
pY C

r Y C

δa

Y C

qz C

pl C

rl C

l C δr

q m

C

p n C

r n C

quantization level [deg]

δa

n C

quantization level [deg]

3210

6420 -0.2

0

0.40.2

-20-40-60

-0.2-0.4

0

0.2

0

0.5

1 0.04

0.030.02

-35-40-45-50 -1

-0.5

0

-1.5

-1

-0.5

0

0 0.5 1 1.5 2

-0.1

-0.05

0

0 0.5 1 1.5 2

Figure 5. Variation in GTM stability and control derivatives with air flow angle resolution (other derivatives)

27
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg/s]

δr

n C

quantization level [deg/s]

-1.2

-1

-0.8

0.3

0.35

0.4

-4-5-6 -1

-0.5

0

-0.2

-0.15

-0.1

-0.1

-0.06-0.08

-2

-1.4-1.6-1.8

-2

-1.2-1.6

0.30.20.1

0 5 10 15 20

-0.1-0.2

0

0 5 10 15 20

Figure 6. Variation in GTM stability and control derivatives with gyroscope resolution

28
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [g]

δr

n C

quantization level [g]

-1.5

-1

-0.5

0

0

0.40.2

-4-5-6

-0.2-0.4-0.6

-0.2

-0.15

-0.1

-0.1

-0.06-0.08

-2

-1.5

-1

-2

-1.2-1.6

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2

-0.2

-0.14-0.16-0.18

0 0.05 0.1 0.15 0.2

Figure 7. Variation in GTM stability and control derivatives with accelerometer resolution

29
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg/s]

δr

n C

quantization level [deg/s]

-1.5

-1

-0.5

0

0

0.60.40.2

-2-4-6

0

-1-2

10

-0.1-0.2

0

-0.1

-0.05

0

-1-2

0

-2.5

-2

-1.5

-1

0

0.30.20.1

0 5 10 15 20

-0.2

-0.15

-0.1

-0.05

0 5 10 15 20

Figure 8. Variation in GTM stability and control derivatives with gyroscope resolution and air flow angle reconstruction

30
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [g]

δr

n C

quantization level [g]

-1-2

10

0

0.40.2

-4-5-6 -1

-0.5

0

-0.1-0.2

0

-0.1

-0.05

0

-2

-1.5

-1

-2

-1.2-1.6

0

0.30.20.1

0 0.05 0.1 0.15 0.2

-0.2

-0.14-0.16-0.18

0 0.05 0.1 0.15 0.2

Figure 9. Variation in GTM stability and control derivatives with accelerometer resolution and air flow angle reconstruction

31
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [lbf/ft2

]

δr

n C

quantization level [lbf/ft2

]

-1.2-1.4

-1

-0.8

0.3

0.35

0.4

0.45

-4-5-6 -1

-0.5

0

-0.2

-0.15

-0.1

-0.1

-0.06-0.08

-2

-1.4-1.6-1.8

-2.4

-2

-1.2-1.6

0.15

0.2

0.25

0.3

0 2 4 6 8 10

-0.2

-0.14-0.16-0.18

0 2 4 6 8 10

Figure 10. Variation in GTM stability and control derivatives with dynamic pressure resolution

32
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg]

δr

n C

quantization level [deg]

-1.2

-1

-0.8

0.25

0.3

0.35

0.4

-4-5-6

-0.4-0.5-0.6

-0.2

-0.15

-0.1

-0.1

-0.05

0

-2

-1.5

-1

-2

-1.2-1.6

0.15

0.2

0.25

0 0.5 1 1.5 2

-0.2

-0.15

-0.1

0 0.5 1 1.5 2

Figure 11. Variation in GTM stability and control derivatives with control surface potentiometer resolution

33
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

error [%]

δr

n C

error [%]

-1.5

-1

-0.5 0.5

0.40.30.2

-3-4-5-6

-0.2-0.4-0.6

-0.2

-0.15

-0.1

-0.1

-0.04-0.06-0.08

-2

-1.5

-1

-2.4

-2

-1.2-1.6

0.15

0.2

0.25

0.3

0 5 10 15 20

-0.25

-0.2

-0.15

-0.1

0 5 10 15 20

Figure 12. Variation in GTM stability and control derivatives with mass and inertia resolution

34
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

error [%]

δr

n C

error [%]

-1.5

-1

-0.5 0.5

0.40.30.2

-3-4-5-6

-0.2-0.4-0.6

-0.2

-0.15

-0.1

-0.1

-0.04-0.06-0.08

-2

-1.5

-1

-2.4

-2

-1.2-1.6

0.15

0.2

0.25

0 5 10 15 20

-0.25

-0.2

-0.15

-0.1

0 5 10 15 20

Figure 13. Variation in GTM stability and control derivatives with mass and inertia resolution and negligible

Ixz

35
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

error [%]

δr

n C

error [%]

-1.5

-1

-0.5

0.3

0.35

0.4

-4-5-6

-0.4-0.5-0.6

-0.2

-0.15

-0.1

-0.1

-0.06-0.08

-2

-1.5

-1

-2.4

-2

-1.2-1.6

0.15

0.2

0.25

0 2 4 6 8 10

-0.2

-0.15

-0.1

0 2 4 6 8 10

Figure 14. Variation in GTM stability and control derivatives with aircraft geometry resolution

36
=0 α

0.0 deg

Cmα

=0 α

2.0 deg

=0 α

4.0 deg

=0 α

6.0 deg

=0 α

8.0 deg

=0 α

10 deg

resolution [deg/s]

Cmq

resolution [deg/s]

Cmδe

resolution [deg/s]

-2

-1.5

-1-2

-1.5

-1-2

-1.5

-1-2

-1.5

-1-2

-1.5

-1-2

-1.5

-1

0 5 10 15 20

-20-40

0

-20-40

0

-20-40

0

-20-40

0

-20-40

0

-20-40

0

0 5 10 15 20

-2

-1.5

-1

-0.5

-2

-1.5

-1

-0.5

-2

-1.5

-1

-0.5

-2

-1.5

-1

-0.5

-2

-1.5

-1

-0.5

-2

-1.5

-1

-0.5

0 5 10 15 20

Figure 15. Variation of GTM pitching moment derivatives with varying gyroscope resolution for several trim conditions

37
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg]

δr

n C

quantization level [deg]

-1.5

-1

-0.5

0 0.5

0.40.3

-3-4-5-6 -1

-0.5

0

-0.1-0.2

0

-0.1

-0.05

0

-2

-1.5

-1

-2

-1.2-1.6

0

0.30.20.1

0 0.5 1 1.5 2

-0.2

-0.14-0.16-0.18

0 0.5 1 1.5 2

Figure 16. Variation in GTM stability and control derivatives with air flow angle resolution using equation-error parameter estimation in thetime domain

38
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg]

δr

n C

quantization level [deg]

-2

-1.5

-1

-0.5

0

0.25

0.3

0.35

0.4

0.45

-4-5-6 -1

-0.5

0

-0.1-0.2-0.3

0

-0.15

-0.1

-0.05

0

-2

-1.5

-1

-2

-1.2-1.6

0

0.40.30.20.1

0 0.5 1 1.5 2

-0.2

-0.14-0.16-0.18

0 0.5 1 1.5 2

Figure 17. Variation in GTM stability and control derivatives with air flow angle resolution using output-error parameter estimation in thetime domain

39
β

Y C

δr

Y C

αZ C

δe

Z C

l C β

l C δa

α m

C

δe

m

C

β

n C

quantization level [deg]

δr

n C

quantization level [deg]

-1.5

-1

-0.5

0

0.1

0.15

0.2

-4

-3.5

-3

-1

-0.5

0

-0.1-0.2

0

-0.2

-0.15

-0.1

-0.05

-0.2-0.3-0.4

-0.4-0.6-0.8

0.30.20.1

0 0.5 1 1.5 2

-0.12

-0.1

-0.08

0 0.5 1 1.5 2

Figure 18. Variation in F-16 stability and control derivatives with air flow angle resolution

40
REPORT DOCUMENTATION PAGE Form Approved

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2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Dependence of Dynamic Modeling Accuracy on Sensor Measurements, 

Mass Properties, and Aircraft Geometry 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Grauer, Jared A.; Morelli, Eugene A.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

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The NASA Generic Transport Model (GTM) nonlinear simulation was used to investigate the effects of errors in sensor 

measurements, mass properties, and aircraft geometry on the accuracy of identified parameters in mathematical models 

describing the flight dynamics and determined from flight data. Measurements from a typical flight condition and system 

identification maneuver were systematically and progressively deteriorated by introducing noise, resolution errors, and bias 

errors. The data were then used to estimate nondimensional stability and control derivatives within a Monte Carlo simulation. 

Based on these results, recommendations are provided for maximum allowable errors in sensor measurements, mass 

properties, and aircraft geometry to achieve desired levels of dynamic modeling accuracy. Results using additional flight 

conditions and parameter estimation methods, as well as a nonlinear flight simulation of the General Dynamics F-16 aircraft, 

were compared with these recommendations.

15. SUBJECT TERMS

Generic Transport Model; Measurement accuracy; System identification

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