Determining Aircraft Moments of Inertia from Flight Test Data

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Determining Aircraft Moments of Inertia

from Flight Test Data

Eugene A. Morelli NASA Langley Research Center, Hampton, Virginia, 236 66 https://doi.org/10.2514/1.G00607 2 Flight test maneuver s and dynamic modeling techniqu es were developed for determining aircraft mome nts of inertia from flight test data. Full nonlinear rigid - body rotational equations of motion were used in the analysis, with a erodynamic moment dependencies modeled by linear expansion s in the aircraft states and controls . A erodynamic parameters were estimated simultaneously with inertia parameters using equation - error modeling applied to flight test data from maneuver s designed specifically for this problem. The approach was demonstrate d using a nonlinear F - 16 simulation, then applied to a remotely - piloted subscale aircraft flight test. E rror s in the a ircraft moment of inertia parameter s determined from simulated F - 16 flight test data w ere less than 6 % compared to the true values in the simulation . Flight test results for the subscale aircraft were within 6 % of ground - test values obtained using the same aircraft .

Nomenclature b = wing span, ft = wing mean aerodynamic chord, ft c C , C ,C = nondimensional aerodynamic roll, pitch, and yaw moment coefficients l m n I = propulsion system rotational moment of inertia, slug - ft p Presented as Paper 202 1 - 1642 at the 2021 AIAA SciTech Forum, Virtual Event, January 11 - 21 , 202 1 Research Engineer, Dynamic Systems and Control Branch, AIAA Associate Fellow I , I , I , I = moments of inertia, slug - ft x y z xz L, M , N = aerodynamic roll, pitch, and yaw moments, ft - lbf m = aircraft mass, slug p, q,r = body - axis rol l, pitch, and yaw angular rates, rad/s or deg/s 2 2 = body - axis rol l, pitch, and yaw angul ar accelerations, rad/s or deg/s p, q, r = dynamic pressure, lbf/ft q rms = root mean square S = wing reference area, ft T = maneuver length, s V = true airspeed, ft/s  = angle of attack, rad or deg  = sideslip angle, rad or deg , , ,     = stabilator, elevator, aileron , and rudder deflections, rad or deg s e a r  = prop ulsion system rotational speed , rad/s p subscripts cg = center of gravity o = reference value or b ias term l , r = left, right I. Introduction TABILITY and control flight testing for fixed - wing aircraft is typically focused on modeling nondimensional

S

aerodynamic force s and moments as a function of explanatory variables such as angle of attack, sideslip angle , body - axis angular rates, and control surface deflections. The nondimensionalization involves dynamic pressure, which is easily measured in flight, and mass/inertia properties [1]. Mass/inertia properties can be obtained from ground tests [ 2 - 17 ] or by carefully constructing a c omputer - a ided d esign (CAD) model of the aircraft [ 18 - 21 ] , along with careful accounting for fuel weight and load ing . D etermin ing aircraft mass on the ground is simple , but ground testing for determin ing moments of inertia is both costly and time - consuming, and can sometimes result in damage to the air craft . In addition, any substantial change to the aircraft configuration incurs changes in the inertia properties, which r equ ires either an adjustment to previous ground test results or a repeat of the ground tests. Using a CAD model requires detailed knowledge of the mass and location of all aircraft components , which also must be carefully updated for any aircraft configuratio n change s .

Previous work ha s addressed t he problem of determining inertia properties from flight data for quadrotors [ 22 - 23 ] and spacecraft [ 24 - 27 ], assuming that the applied moments are known. For fixed - wing aircraft flying in the atmosphere, the problem is more complicated, because the applied moments from aerodynamics cannot be treated as known.

In this work, novel flight test maneuver s and dynamic modeling technique s were develo ped to determine the moments of inertia for symmetric fixed - wing aircraft from flight test data alone. The approach requires an instrumented aircraft capable of controlled flight test maneuver s with high angular rates . Unmanned rapid - prototype aircraft or subscale research aircraft can satisfy these requirements. For other flight testing, the method might be used to validate or corroborate ground test results, or to determine the change s in moments of inertia associated with aircraft configuration changes.

The general idea is to fly a maneuver at low nominal angle of attack with small excursions in angle of attack and sideslip angle induced by low - amplitude perturbations of the control surfaces , while simultaneously invoking high - amplitude body - axis angula r rates. This keeps the aircraft aerodynamic dependencies approximately linear, while enhancing the nonlinear terms in the rotational dynamic equations of motion that involv e the aircraft moments of inertia. Because the terms associated with the aircraft m oments of inertia are nonlinear in the angular rates and different from aerodynamic dependencies for small perturbations in angle of attack and sideslip angle at low nominal angles of attack, both the linear aerodynamic parameters and the moment of inertia parameters can be estimated accurately and simultaneously using equation - error parameter estimation applied to the flight data.

The next section explains the method, including the equations , modeling computations , and the flight test maneuver design . In Section III, the F - 16 nonlinear simulation and the E1 subscale aircraft are described. Section IV demonstrates an application of the method using simulated flight data from the F - 16 nonlinear simulation with realistic m easurement noise and known moments of inertia. In Section V, the method is applied to flight test data from the E1 subscale aircraft flown by a pilot on the ground using conventional radio control and automated excitation inputs applied to the control surf aces . Aircraft moments of inertia from ground testing for this aircraft we re used for comparison with the flight test results. Section VI provides a discussion of the methods and results and Section VII contains conclusions.

All of the flight test maneuv er design, flight simulation, data analysis , and modeling tasks for this work were ® done using a software toolbox written in MATLAB called System IDentification Programs for AirCraft ( SIDPAC ) [1 , 28 ] . SIDPAC was developed at NASA Langley and has been applied successfully to a wide variety of flight test and wind tunnel experiments . SIDPAC has been used at more than 100 organizations worldwide to solve aircraft system identification problems [ 29 ].

II. Meth od D etermining aircraft moments of inertia from flight test data involve s appropriate equations of motion, modeling assumptions and techniques for estimat ing the unknown model parameters, a long with specific flight test maneuvers to generate suitable flig ht data.

A. Equations of Motion The nonlinear equation s of motion for the rigid - body rotational dynamics of a symmetric fixed - wing aircraft with thrust acting along the x body axis are [1]: I p I r L I I qr I pq − = + − + (1)

( )

x xz y z xz 2 2 I q M I I pr I r p I r  = + − + − + (2)

( )

( )

y z x xz p p I r I p N I I pq I qr I q  − = + − − − (3)

( )

z xz x y xz p p For a conventional airplane, if the flight test maneuver is conducted s o that the aerodynamic dependencies can be modeled using linear expansions in the aircraft states and controls, pb rb   L qSb C C C C C C    = + + + + + (4) l l l l l a l r   o p r    a r 2 2 V V   qc   M qSc C C C C   = + + + (5) m m m m e   o q   e 2 V   pb rb   N qSb C C C C C C    = + + + + + (6) n n n n n a n r   o p r    a r 2 2 V V   Combining Eqs. (1) - (6) gives the rotational equations of motion with nonlinear dynamics and linear aerodynamics, pb rb   I p I r qSb C C C C C C I I qr I pq    − = + + + + + + − + (7)

( )

x xz l l l l l a l r y z xz   o p r    a r 2 2 V V   qc   2 2 I q qSc C C C C I I pr I r p I r    = + + + + − + − + (8)

( )

( )

y m m m m e z x xz p p   o q   e 2 V   pb rb   (9) I r I p qSb C C C C C C I I pq I qr I q     − = + + + + + + − − −

( )

z xz n n n n n a n r x y xz p p   o p r    a r 2 2 V V   Rearranging , qSb pb rb   p C C C C C C c qr c r pq    = + + + + + + + +

( ) (1 0 )

1 2 l l l l l a l r   o p r    a r 2 2 I V V   x I qS c qc p   2 2 q C C C C c pr c r p r    = + + + + + − + (1 1 )

( )

3 4 m m m m e p   o q   e 2 I V I   y y I qSb pb rb p   r C C C C C C c pq c p qr q     = + + + + + + + − −

( ) (1 2 )

5 6 n n n n n a n r p   o p r    a r 2 2 I V V I   z z where c I I I = − c I I = (1 3 a)

( )

1 y z x 2 xz x c I I I = − c I I = (1 3 b)

( )

3 z x y 4 xz y c I I I = − (1 3 c) c I I =

( )

5 x y z 6 xz z The aerodynamic bias terms C , C ,C must be retained in Eqs. (10) - (12), because the nonlinear equations of l m n o o o motion are being used. Propulsion system rotational inertia I can be determined from simple ground tests or from p manufacturer data, and the propulsion system rotational speed  can be measured, so that the gyroscopic terms p involving the propulsion system angular momentum I  can be treated as known. The propulsion terms can also p p  be made small by executing the flight test maneuver at idle power to reduce . F or most fixed - wing aircraft, the p ratios I I and I I will be small , so that the propulsion terms can be negligible compared to the other terms in p y p z the equation . A nother approach is to estimate the propulsion system inertia I as a n additional unknown parameter, p which can be do ne when the propulsion system rotational speed  is measured and varied substantially during the p maneuver.

The re are six inertia constants in Eq . (1 3 ), but only four inertia tensor elements for a symmetric aircraft, namely and I , I , I , I . Consequently, only four of the inertia constants in Eq. (1 3 ) need to be estimated in order to x y z xz obtain values for and I , I , I , I .

x y z xz B. Model ing If the nominal angle of attack during the flight test maneuver is low but the angular rates are high, then Eqs. (1 0 ) - (1 2 ) can be used to estimate both the aerodynamic parameters and the inertia parameter s. Note that the nonlinear angular rate and angular acceleration terms associated with the inertia parameter s are typically not necessary for aerodynamic modeling at low nominal angle s of attack, so that these terms can be associated solely with the inertia effects. All of the inertia parameters in the pitch equation (1 1 ) and the yaw equation (1 2 ) are multiplied by the roll rate p or the roll acceleration p , and the four inertia parameter s in Eqs. (1 1 ) - (1 2 ) involve all four of the inertia tensor e lements. In contrast, the roll equation has the nonlinear term qr , which is difficult to make large while maintaining a low nominal angle of attack. Furthermore, t he rolling motion of the aircraft needed for sufficient amplitude of the inertial terms in the pitch and yaw equations leads to large aerodynamic terms in the roll equation . In that case , aerodynamic terms dominate t he inertia terms in the roll equation , w hich makes the inertia terms in the roll equation difficult to determine accurately . A good approach is to use flight test maneuvers with large - amplitude changes in roll rate, then analyze the data using only the pitch and yaw moment equations.

Because t he airspeed V and the dynamic pressure q change significantly for most high - amplitude maneuvers, those depende ncies must be retained in the aerodynamic modeling. E quations (1 0 ) - (1 2 ) can be written as p r p q L q L q L q L q L q L c qr c r pq    = + + + + + + + + (1 4 )

( )

1 2 o p r a r    a r V V I q p 2 2 q q M q M q M q M c pr c r p r    = + + + + + − + (1 5 )

( ) 3 4 o q e p  

e V I y I p r p r q N q N q N q N q N q N c pq c p qr q     = + + + + + + + − − (1 6 )

( )

5 6 o p r a r p    a r V V I z where 2 2 Sb Sb Sb Sb Sb Sb L C L C L C L C L C L C = = = = = = ( 17 a ) o l l p l r l l l    o p r a r    a r 2 2 I I I I I I x x x x x x S c S c S c S c M C M C M C M C = = = = ( 17 b ) o m m q m m   o q e   e 2 I I I I y y y y 2 2 Sb Sb Sb Sb Sb Sb N C N C N C N C N C N C = = = = = = ( 17 c ) o n n p n r n n n    o p r a r    a r 2 2 I I I I I I z z z z z z Although the unknown parameters specified in Eq. ( 17 ) involve both aerodynamic s and inertia, these quantities c an be considered constant for a flight test maneuver executed at low nominal angle of attack. The definitions in Eq. ( 17 ) are not the usual definitions of dimensional derivatives, because dynamic pressure and airspeed are specified separately in Eqs. (1 4 ) - (1 6 ). E stimates of the unknown parameters defined in Eq. ( 17 ) will not be useful directly for aerodynamic modeling, but rather are used as a means toward t he goal of estimating the inertia parameters . T he inertia parameters are determined from a flight test maneuver designed to make the nonlinear inertia terms significant. T he resulting inertia tensor element value s can then be used to compute nondimensional aerodynamic model parameters using Eq. ( 17 ), or in separate analyses of standard small - amplitude maneuver s , where the nonlinear inertial terms are assumed negligible because of low angular rates.

Equations (1 4 ) - (1 6 ) are coupled nonlinear equations with unknown parameters characterizing the linear aerodynamic dependencies and the moments of inertia. Using only Eqs. (1 5 ) and (1 6 ) for the modeling, and assuming that the flight test maneuver is executed at idle power to make the propulsion sys tem gyroscopic terms negligible, the vector of unknown parameters θ is T   M M M M c c N N N N N N c c  θ (18) 3 4 5 6 o q o p r      e a r   If the roll equation (1 4 ) is omitted from the analysis , then the flight data for roll rate p and roll acceleration p can be substituted as measured values in Eqs. (1 5 ) and (1 6 ). This helps the modeling by reducing the number of unknown parameters without compromising the goal of accurately estimating at least four inertia parameters , which a re and c ,c ,c , c in Eqs. (1 5 ) and (1 6 ).

3 4 5 6 The resulting modeling problem ca n be solved using equation - error parameter estimation with nonlinear model terms in either the time domain [1] or the frequency domain [1, 30 ]. Because the model structure for the problem is fixed, real - time parameter estimation methods [1, 30 ] can be used a s well. T he problem can also be formulated as a nonlinear output - error parameter estimation problem in the time domain, which can be solved using standard optimization routines [1 , 31 ] .

In this work, equation - error in the time domain was used with smoothed explanatory variable data to obtain accurate and unbiased parameter estimates [1, 32 ]. This approach allows easy incorporation of data from multiple maneuvers and the capability to selec tively include or exclude data points that are not necessarily contiguous in time. These features are helpful for improving relevant data information content by including data from multiple maneuvers at low nominal angles of attack where a linear aerodynam ic model is valid.

The equation - error approach also allows modeling using one equation at a time. For example, the equation - error parameter estimation problem for the pitching moment equation (15) with N data points is (19) = + z X θ  wh ere T

( ) ( ) ( ) 1 2 vector of output measurements q q q N = =   z

  1 q   ( ) 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 q q q q p r r p   − e   1 V ( )     2 q ( ) 2 2   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 q q q q p r r p   − e 2 V   ( ) matrix of modeling function vectors = = X       ( ) q N   2 2 q N q N N q N q N N p N r N r N p N   − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e   ( ) V N   T   vector of unknown parameters M M M M c c  = θ 3 4 o q   e   T 1 2 vector of equation errors N    = = 

( ) ( ) ( )  

  The matrix X is assembled using measured data, with each column representing a modeling function, also called a regressor. The measured output data z is computed as a smoothed numerical derivative of the measured angular rates [1 , 32 ]. The best estimate of θ in a least - squares sense comes from minimizing the sum of squared differences between the measured output z and the model output = y X θ , T

( ) ( ) ( ) J = − − θ z Xθ z Xθ ( 20 )

The least - squares solution for the unknown parameter vector is [1] θ 1 − T T ˆ = θ X X X z ( 21 )

( )

and the estimated model output is ˆ ˆ = y X θ ( 22 ) The estimated parameter covariance matrix is computed from [1] 1 − 2 T ˆ ˆ   Cov C  =  θ X X 1 2 i, j , , ,n = ( 23 a)

( )

( ) ij p

  T ˆ ˆ − − z y z y

( ) ( )

ˆ  = ( 23 b) N n −

( )

p where n is the number of unknown parameters , and 6 n = for this example. Modifications of Eq. (23) are p p ˆ required when the residuals ( ) z y − are colored [1 ,32 ]. The standard errors of the estimated parameters are given by the square root of the diagonal elements of th e covariance matrix, ˆ s C  = 1 2 j , , ,n = ( 2 4)

( ) j jj p

E quation - error modeling for the yawing moment Eq. (16) is similar. The inertia parameters and c ,c ,c , c in 3 4 5 6 Eqs. (15) and (16) are linear model parameters in the equation - error formulation, because the parameter s appear linearly in the equations. If instead the unknown inertia parameters are changed to the inertia tensor elements and I , I , I , I , then the se unknown inertia parameters appear nonlinearly in the equations, which requires a x y z xz nonlinear optimizer for the solution [1]. Either of these approaches can be readily implemented using SIDPAC.

In Eqs. ( 1 4 ) - ( 1 6 ), the aerodynamic parameters are multiplied by terms that are linear in sideslip angle, angle of attack, angular rates, and control surface deflection s , whereas the moment of inertia parameters are multiplied by angular accelerations and nonlinear functions of the angular rate s . This distinction is the basis for the ability to identify aerodynamic parameters and the moment of inertia parameters simultaneously. However, to do so requires a flight test maneuver for which the aerodynamic dependencie s can be considered linear while the angular rates and angular a ccelerations change significantly. F light test maneuver s with these characteristics are described next.

C. Flight Test Maneuver Design The flight test maneuver design ha s two objectives: 1) achieve large - amplitude angular rates, so that the nonlinear terms associated with the inertia parameters ar e significant, and 2) excite the aerodynamic explanatory variables in an uncorrelated way at low nominal angle of attack, to ena ble accurate linear aerodynamic model parameter estimation. Th ese objectives can be achieved by flying a large - amplitude maneuver with orthogonal optimized multisine excitations active throughout the maneuver.

1. Large - Amplitude Maneuver Relatively large an gular rates can be achieved at low nominal angle of attack by roll ing about the velocity vector [33] in to descending turns. As discussed earlier, the nonlinear inertia terms in the pitch and yaw equations all involve roll rate or roll acceleration. Velocit y - vector r olls in to descending turns with approximately constant nominal angle of attack were found to be effective for raising the roll rate and roll acceleration amplitudes with simultaneous pitch and yaw rates, and are reasonable to fly. Following alter nating descending turns, a gradual pullup can be used to arrest the aircraft descent and increase the pitch rate. Variations of this sequence or o ther flight test maneuvers of this kind c an also be acceptable , as will be discussed in the application exampl es.

For l inear aerodynamic parameter estimation , uncorrelated excitations of the explanatory variables about the nominal flight condition are required . This excitation was provided by automated orthogonal optimized multisine inputs [1, 30 , 34 - 37 ] , describe d next . These perturbation input s are balanced about zero with minimized amplitude excursions , so that the aircraft maintains its nominal trajectory, but the aerodynamic explanatory variables are excited in an uncorrelated manner, enabling accurate linear aerodynamic parameter estimation. The se inputs also excit ed the angular rates, so that the nonlinear terms associated with the inertia parameters were further enhanced .

Automated orthogonal optimized mult isine inputs were applied to multiple control surfaces simultaneously throughout the large - amplitude maneuver s . These automated excitation inputs were called Programmed Test Inputs ( PTIs ) for both the F - 16 nonlinear simulation and the E1 flight tests.

2. Ort hogonal Optimized Multisine Inputs The general idea for the small - amplitude excitations ( PTIs ) is to move the aircraft control surfaces in a manner that decorrelates the explanatory variables, usin g perturbation inputs with wideband frequency content encompassing the expected modal frequencies of the aircraft dynamic response. The excitations are implemented by summing designed perturbation inputs with the actuator commands from the pilot and/or feedback control system , just before the limiting on actuator command rate s and position s . This implementation is important for achieving the required excitation and low correlations in the explanatory variables.

Each designed perturbation input is a sum of sinusoids with unique harmonic frequencies, optimized phase angles , and specified power distribution. The wideband frequency content of the inputs is important because there is naturally some uncertainty as to what the modal frequencies are for the aircraft in flight, and wide band inputs provide robustness to that uncertainty. Multiple inputs are designed to be mutually orthogonal in both the time domain and the frequency domain simultaneously, and are designed for high data information content in all axes, while minimizing exc ursions from the nominal flight condition . The mutual orthogonality of the inputs allows simultaneous application of multiple inputs, which reduces the required excitation time for a given amount of input energy, or equivalently, increases the amount of input energy injected into the dynamic syst em over a given time period. T he inputs provide continuous, effective, multi - axis excitation as the aircraft flies along a nominal flight trajectory .

Each perturbation input u applied to the j th control surface is a sum of harmon ic sinusoids with unique j frequencies and individual phase angles  , jk 2 t k    sin u A  = + (25) 1 2 j , , ,n =

j jk jk    i

T   1 2 k , , ,M    where M is the total number of available harmonic frequencies, T is the time length of the excitation, A is the jk t amplitude for the k th sinusoidal component , and is the time vector. E ach of the n input s is the sum of selected i components from the pool of M harmonic sinusoids with frequencies 2 1 2 k T , k , , ,M   = = , and k 2 M T   = represents the upper limit of the frequency band for the excitation. The interval ,   rad/s

 

M 1 M specifies the frequency range where the aircraft dynamics are expected to lie. Each u in Eq. (25) is a PTI applied j to an individual control surface.

The mutual orthogonality of the PTIs in the time domain comes from the fact that each input is composed of harmonic sinusoids with the same base period T but unique harmonic frequencies. Orthogonality in the frequency do main comes from using unique frequencies for the component sinusoids in each multisine input. Both orthogonalit y properties exist simultaneously for all in puts . The mutual orthogonality of the inputs helps the dynamic modeling by completely decorrelating the inputs, which improves the accuracy of control effectiveness estimates and reduces the correlations among the other explanatory variables as well. This proper ty also means that the PTIs for every control surface can be applied simultaneously , which produces high information content in the data very efficiently.

If the phase angles  in Eq. ( 25 ) were chosen at random on the interval ( ,   − rad, then in general, the



jk various harmonic components would add together at some points to produce a multisine input u with relatively j large amplitude excursions. This is undesirable because such inputs can move the aircraft too far away from the nominal flight condition . To prevent this, the phase angles  for each of the selected harmonic components are jk optimized to minimize the relative peak factor RPF for each input, defined by max min u u −

( ) ( )

j j RPF u = 1 2 j , , ,n = (26)

( )

j i 2 2 rms u

( )

j Relative peak factor is a measure of the efficiency of an input for dynamic modeling purposes and is computed as the amplitude range of the input divided by a measure o f the input energy. Low values of relative peak factor are desirable for highly efficient and effective modeling because the objective is to excite the aircraft with good input energy over a variety of frequencies while minimizing the input amplitudes in t he time domain, to avoid driving the aircraft too far away from the reference condition. For each multisine input u in Eq. ( 2 5 ), minimum RPF is j  achieved by adjusting the phase angles for each individual sinusoidal component of the input. The resulting jk optimization problem is multi - dimensional and non - convex; however, a simplex algorithm can be applied to find a solution. The orthogonality of the inputs is unaffected by the values chos en for the phase angles  [1 ,34 - 36 ].

jk The integers k specifying the harmonic frequencies for the j th input u are selected to be unique to that input, j but are not necessarily consec utive. A good approach for multiple inputs is to assign the harmonic frequencies to the input s alternately. This is illustrated in Fig. 1 for the PTI design on the E1 aircraft. There are 4 inputs in this case : left aileron, right aileron, elevator , and rudder . A total of 44 frequencies ( ) 44 M = were used with excitation time period 40 s T = over the frequency band [0.05,1.65] Hz . The harmonic frequencies were interleaved among the four inputs to achieve wideband frequency content in each input, provide robustness in the excitation , and enable accurate estimates of individual control surface effectiveness. Because each input has wideband frequency content, the same input design can be applied at various flight conditions, which simplifies the excitation strategy and reduces flight computer memory requirements. Figure 2 shows time series for the PTIs designed using the frequency content depicted in Fig. 1 .

 a l deg  a r deg P k  e deg  r deg Figure 1 . Orthogonal optimized multisine input spectra Figure 2. Orthogonal optimized multisine inputs The sinusoidal components in Eq. (2 5 ) can be assigned arbitrary fractions of the total power in the multisine input, to emphasize the excitation at selected frequencies. This is implemented by choosing sinusoidal compon ent amplitudes as A A P = (27) jk j jk where A is the amplitude of the multisine input u , and P is the power fraction for the k th sinusoidal j j jk u component of . The power fractions for the sinusoidal components in each multisine input must sum to 1, j 1 P = (28)

jk 

k To achieve a uniform power distribution, as shown in Fig. 1, A are selected as jk A j A = (29) jk M j where M is the number of sinusoidal components included in the summation of Eq. (2 5 ) for u , and A is the j j j u A amplitude of the multisine input . With uniform power distribution, selection of the reduces to selecting a j jk single value for the input amplitude A . Each input u can have arbitrary amplitude A , subject to practical flight j j j test and modeling constraints. The power spectra shown in Fig. 1 are power fractions ( P ), so that the individual jk control surface amplitudes (3 deg for all control surfaces , see Fig. 2) are excluded. SIDPAC program mkmsswp.m was used to design the orthogonal optimized multisine input s ( PTIs ) used in this work.

The PTI design shown in Fig s . 1 and 2 was used for the E1 flight test s . Because the PTIs are sums of harmonic sinusoids with a common base period T , they are periodic for the excitation period T , so that the PTIs can be applied repeatedly without any discontinuities in magnitude or slope. The PTI design has minimum RPF and various frequencies and phase angles, which keep s the aircraft response close to the nominal flight trajectory and produces dynamic responses similar to flight in light - to - moderate turbulence. The aircraft response also stays near the nominal flight trajectory because each perturbation input is a sum of harmonic sinusoids, which are all balanced about zero amplitude (equal area above and below zero). The result is rich , dyna mic , multi - axis response about the nominal flight trajectory. In practice, pilot inputs and/or automated guidance and control act to spoil the orthogonality (zero pairwise correlations) of the PTI design. However, good modeling results require only low cor relations, not zero correlations, so that the slightly correlated inputs that result from applying orthogonal PTIs with pilot inputs and/or automated guidance and control active still work very well in practice.

For flight test situations where an onboar d automated excitation system is not available to implement orthogonal optimize d multisine inputs, previous flight tests have demonstrated that a test pilot can implement effective multi - axis perturbation inputs with low correlations during large - amplitude maneuvering [ 38 ].

III. Aircraft A. F - 16 Nonlinear Simulation The F - 16 is a single - seat, multi role fighter with a blended wing - body and a cropped delta wing planform with leading - edge sweep of 40 deg. Thrust is provided by one General Electric F110 - GE - 100 or Pratt & Whitney F100 - PW - 220 afterburning turbofan engine mounted in the rear fuselage. Figure 3 is a photograph of the F - 16 in flight.

Aircraft geometry and nominal mass /inertia properties are given in Table 1.

The F - 16 nonlinear simulation has controls for throttle  , stabilator  , aileron  , and rudder  . Speed th s a r brake and flaps were assumed fixed at zero deflection.

Table 1. Aircraft geometry and mass/inertia properties Property F - 16 E1 , ft c 11.32 1.97 b , ft 30 10.17 , ft S 300 19.26 0.35 c 3.012 x , ft o 0 0 y , ft o 0 0 z , ft o x 0.25 c 3.027 , ft cg y 0 0.028 , ft cg z , ft 0 −0.248 cg Figure 3 . F - 16 air craft m , slug 637 1.910 Credit : NASA Langley Research Center I , slug - ft 9,496 2.964 x I , slug - ft 55,814 8.776 y 63,100 11.716 I , slug - ft z I , slug - ft 982 0.750 xz Nondimensional nonlinear aerodynamic force and moment coefficient data were derived from a low - speed static wind - tunnel test and a dynamic forced oscillation wind - tunnel test, both conducted using a 16% scale model of the F - 16. The aerodynamic database applies to the F - 16 flown out of ground effect, with landing gear retracted and no external stores , over a wide range of angle of attack and sideslip angle .

® The F - 16 nonlinear simulation was programmed completely in MAT LAB . Full nonlinear equations of motion, including turbine engine gyroscopic effects, were used. Complete details on the F - 16 nonlinear simulation can be found in Ref. [1], Appendix D.

B. E1 Aircraft A subscale aircraft designated E1 was used for flight te sting. The E1 aircraft is a commercially available 40% scale Extra 330 SC remotely piloted fixed - wing airplane, shown in Fig. 4 . E1 is powered by an electric motor driving a fixed - pitch tractor propeller. Control surfaces are conventional ailerons and trai ling - edge flaps on the wings, along with a conventional rudder and split elevator. Aircraft geometry and mass /inertia properties are given in Table 1. More information on the E1 aircraft, flight test instrumentation, and flight test operations can be found in Ref s . [ 39 , 40 ].

The flight computer on the E1 aircraft has the capability to inject automated control surface perturbation s, called Programmed Test Inputs ( PTIs ), to excite the aircraft dynamic response and decorrelate the aircraft states and controls, thereby generat ing flight data with high information content for dynamic modeling .

Figure 4 . E1 aircraft Credit : NASA Langley Research Center Unique PTIs were applied to each control surface simultaneously, by summing the PTIs with commands from the pilot, just before the limiting on control surface actuator command rates and positions. F light data used in this work were collected at 50 Hz with the PTIs ac tive and a radio - control pilot on the ground flying the aircraft through large - amplitude maneuvers.

IV. F - 16 Nonlinear Simulation Example The F - 16 nonlinear simulation described earlier was used to demonstrate the method for estimating moments of inertia fro m flight test data. The flight test maneuver was composed of piloted alternating velocity - vector roll entries into descending turns followed by a gradual pullup to arrest the descent while maintaining low nominal angle of attack , with automated orthogonal optimized multisine input excitations ( PTIs ) active throughout the maneuver. The maneuver was flown at idle power, which made the gyroscopic terms from the propulsion system angular momentum negligible. Figure 5 shows the aircraft control surface deflectio n s and response data. Gaussian white noise was added to the simulation outputs, with noise magnitudes chosen to achieve signal - to - noise ratio of approximately 2 0 for each aircraft response. Note that the angle of attack and sideslip angle varied over a rel atively small range near their nominal values while the angular rates exhibited large amplitudes , particularly the roll rate .

This was done to keep the aerodynamic dependencies approximately linear while making the nonlinear terms associated with the inert ia parameters too large to ignore in the modeling .

Equation - error parameter estimation in the time domain was applied to the simulated flight test data, as described earlier, using SIDPAC program lesq.m . Angular accelerations were computed by applying global Fourier smoothing with SIDPAC program smoo.m , followed by numerical differentiation using SIDPAC program deriv.m . Global Fourier smoothing was also applied to the explanatory variable time series dat a to avoid parameter estimate bias errors that occur when the modeling functions are noisy [1, 32 ]. The aerodynamic and inertia parameters were estimated in the pitch moment equation (1 5 ) and the yaw moment equation (1 6 ), analyzed individually. A comparison of the inertia parameter estimates from simulated flight test data with the known inertia parameters for the F - 16 nonlinear simulation is shown in Table 2. The inertia parameters estimated from simulated flight test data are within 6 % of the true v alues , with the mean absolute error less than 4 % . Standard errors given in Table 2 were corrected for colored residuals using SIDPAC program r_colores.m [1] . A ll true values of the inertia parameters were within ± 2 standard error s of the values esti mated from simulated flight test data , indicating that the estimates were in statistical agreement with the true values . This demonstrates the effectiveness of the approach using realistic simulated flight test data from a large - amplitude maneuver flown by a pilot with PTIs active.

  V deg deg ft/s p q r deg/s deg/s deg/s   q deg deg psf    s a r deg deg deg time, s time, s time, s Figure 5 . F - 16 large - amplitude maneuver data with PTIs active Table 2 . F - 16 Inertia Parameters Parameter True Estimate ± Standard Error Percent Error 0.9604 0.9643 ± 0.0234 0.40 c 0.0176 0.0166 ± 0.0019 – 5.68 c – 0.7340 – 0.7517 ± 0.0189 – 2.41 c 0.0156 0.0147 ± 0.0011 – 5.25 c Figure 6 shows that the model fits to the simulated flight data for angular acceleration in pitch and yaw were 2 2 accurate, with coefficients of determination R equal to 99.9 % and 99.8 % , respectively. The R metric quantifies the model fit to the variation about the mean value for the measured outputs, which are the pitch and yaw angular acceleration data in Fig. 6. More information on the metric can be found in Ref. [1].

R q deg / s r deg / s Figure 6 . Model fit to angular acceleration data for a piloted F - 16 large - amplitude maneuver with PTIs active V. E1 Subscale Aircraft Flight Test The same approach described earlier and applied in the F - 16 nonlinear simulation example was also applied to E1 subscale aircraft flight test data to estimate inertia parameters. Descend ing figure - eight maneuvers were flown at idle power by a pilot on the ground using conventional radio control, while automated orthogonal optimized multisine inputs ( PTIs ) were applied continuously to the elevator, rudder, and individual left and right ailerons.

Rapid 360 - degree v elocity - vector rolls were executed on the straight legs of each descending figure - eight maneuver.

The maneuvers were flown at idle power to minimize the gyroscopic effects f rom the angular momentum of the propulsion system.

Because the E1 flight test maneuvers were flown by a remote pilot on the ground, the pilot had no direct information on the nominal angle of attack . This led to high values of nominal angle of attack and sideslip angle during the rapid 360 - degree velocity - vector rolls , with consequent nonlinear and unsteady aerodynamic effects. To enforce the requirement for linear aerodynamics, the velocity - vector roll data were omitted from the analysis , leaving only th e data from the descending turns. In addition, only data points with angle of attack less than 7 deg were included in the analysis . The se decisions were made based on model structure determination techniques applied to the flight data [1] . T he aerodynamic model required only linear terms , based on statistical modeling metrics used for model structure determination , for data at angle of attack below 7 deg . Because th is data selection process resulted in discarding some data from each maneuver, and to improve the results, data from the descending turn portions of five descending figure - eight maneuvers were combined for the equation - error analysis by simply stacking the selected da ta from each maneuver. Data smoothing was applied to the explanatory variable tim e series prior to the data point selection and data stacking . This was done to obtain accurate and unbiased parameter estimates in the equation - error formulation [1, 32 ] . The PTIs shown in Figs. 1 and 2 w ere applied continuously throughout all of the five m aneuvers. Figure 7 shows the E1 flight trajectory for one of the five descending turn maneuvers with PTIs active throughout the maneuver.

Figure 7 . E1 large - amplitude maneuver trajectory with PTIs active Figure 8 shows the E1 flight test data used for the equation - error parameter estimation, after removing the rapid velocity - vector roll data and prior to the data selection based on angle of attack . The flight data shown are concatenated data from five descending turn maneuvers, for a total of approximately 70 s econds of flight data.

The model fits to E1 flight data for angular acceleration in pitch and yaw are shown in Fig. 9. Coefficient of determination R w as 94.7 % and 94.9 % for the pitch and yaw angular acceleration s , respectively. Table 3 shows the inertia parameters estimated from E1 flight test data , along with values obtained from ground testing using the same E1 aircraft. In this case, the inertia tensor elements were estimated directly, but still using the same equation - error formulation. As discussed earlier, this require s a nonlinear optimizer to obtain the inertia tensor parameter estimates , because the inertia tensor elements appear nonlinearly in the equations . SIDPAC program oe.m was used for this purpose.

The ground test values in Table 3 were determined using a tri - filar torsional pendulum for I and I , an z xz overhead pivot pendulum for I and I , a point - mass correction for a minor elevator repair, and corrections for air x y resistance based on measurements fro m similarly configured airframes. Uncertainty in the ground test results was estimated a t less than ±3 % .

The mean absolute difference in the flight test estimates relative to the ground - test results was less than 4 % , with a maximum absolute percent difference of 5 . 80 % , as shown in the last column of Table 3 . S tandard errors for the flight test results were computed using SIDPAC program m _colores.m to correct the equation - error model parameter uncertainties for colored residuals when the model parameterization is nonlinear, as is the case when estimating inertia tensor elements in the equation - error formulation. The results in Table 3 show that a ll ground - test values of the inertia parameters were within ±2 standard erro rs of the values estimated from E1 flight data, indicating that the flight estimates were in statistical agreement with the ground - test values.

The E1 flight test results demonstrate that the proposed approach is a feasible and accurate method for estima ting moment of inertia parameters directly from flight test data.

  V deg deg ft/s q r p deg/s deg/s deg/s   q deg deg psf    a e r l deg deg deg time, s time, s time, s Figure 8 . E1 large - amplitude maneuver data with PTIs active q deg / s r deg / s Figure 9 . Model fit to angular acceleration data for piloted E1 large - amplitude maneuv ers with PTIs active Table 3 . E1 Inertia Parameters Parameter Ground Test Flight Test Estimate Percent Estimate ± Standard Error Difference 2.964 3.0 53 ± 1.002 3.01 I x 8.776 8. 711 ± 0. 732 – 0. 74 I y I 11.716 1 1 . 036 ± 0. 475 – 5 . 80 z 0.750 0.7 79 ± 0. 107 3 .8 9 I xz VI. Discussion The F - 16 nonlinear simulation and E1 flight test examples demonstrated that inertia parameters can be determined accurately using either linear estimation for and c ,c ,c , c , or nonlinear estimat ion for 3 4 5 6 and I , I , I , I . The latter approach is more direct, in that the results are the inertia tensor elements, and the x y z xz calculated uncertainties apply directly to those quantities. However, it was found that the nonlinear estimation approach r equired more flight data to obtain good accuracy. The equation - error formulation accommodates this requirement , because it is easy to stack data from several maneuvers for one analysis, as was done for the five maneuvers flown on the E1 aircraft. T he linea r estimation approach requires less flight data, as demonstrated by the F - 16 nonlinear simulation example, which used data from a single simulated flight test maneuver. This approach can also be implemented using real - time parameter estimation methods. T he results are estimates of the inertia constants and c ,c ,c , c and associated uncertainties, which must then be converted to inertia tensor elements 3 4 5 6 and I , I , I , I for nondimensional aerodynamic modeling . An analytic expression fo r the transformation from x y z xz and I , I , I , I and c ,c ,c , c to [ i.e., the inverse of Eq. (13) ] could not be found, even with the assistance 3 4 5 6 x y z xz of symbolic mathematics software. However, it was found that a numerical nonlinear solver c ould be applied to find a n accurate solution for the inverse transformation of Eq. (13).

It is likely that a variety of different large - amplitude maneuvers could be used successfully with the proposed method, which might result in even more accurate flight test results. A good maneuver for this problem will produce uncorrelated aerodynamic explanatory variable data at low n ominal angles of attack and high angular rates. G iven the capability to select data points from individual maneuvers and combine data from several maneuvers, the recommended approach is to fly multiple maneuvers to collect the required flight data , then se lect the data from th os e maneuvers that satisfy the requirements of low nominal angle of attack , low correlations among the aerodynamic explanatory variables, and high angular rates.

VII. Conclu sions A novel method for accurately estimating aircraft moments o f inertia directly from flight test data was explained and demonstrated. The approach uses flight test data from maneuver s designed for high angular rates to make the nonlinear inertia terms significant, with simultaneous orthogonal optimized multisine exc itations and low nominal angle of attack for accurate linear aerodynamic parameter estimates . The required equations of motion and modeling techniques were developed and the method was demonstrated using simulated flight data from an F - 16 nonlinear simulat ion. The method was then applied to flight test data from the E1 subscale aircraft, and the flight - estimated moments of inertia were compared with values obtained from ground test ing using the same aircraft .

Results showed that the approach is feasible a nd accurate, but requires an instrumented aircraft that can be flown in maneuvers with high angular rates and an automated onboard excitation system. These requirements can be fulfilled by many subscale aircraft used in research and development, or by rapi d - prototype aircraft. Previous flight tests have demonstrated th at a skilled test pilot can implement effective multi - axis excitations with low correlations during large - amplitude maneuver s , so that it should be possible to use this technique without an onboard automated excitation system . R epeated maneuvers can be flown for improved accura cy and confidence in the results . The equation - error formulation can easily include or exclude data points from multiple maneuvers to satisfy the requirement s of low nominal angle of attack with high angular rates . Applying d ata smoothing to the explanatory variable time series data prior to the data point selection produces parameter estimates that are unbiased and accurate. Results showed that errors in the aircraft moment of inertia parameters determined from simulated F - 16 flight test maneuver data were less than 6 % compared to the true values in the simulation. Flight test results for the E1 subscale aircraft were within 6 % of ground - test val ues obtained using the same aircraft.

A good practical flight test procedure would be to first apply the proposed method to accurately estimate the inertia tensor elements, then proceed with stability and control flight testing using nondimensional aerody namic modeling. The proposed flight test technique is faster than ground - based inertia testing, and the results shown in this work demonstrate that the accuracy is comparable. R isk to the airframe is arguably lower with the proposed flight test method, bec ause fairly simple flight test maneuve rs are required, whereas ground - based inertia testing typically involves swinging or oscillating the suspended aircraft in some way, or mounting the aircraft on an apparatus with springs and pivots. Furthermore, the st atistical uncertainty associated with results of the flight test method can be computed accurately with well - established methods, whereas determining the uncertainty in results from ground - based inertia testing is more complex and difficult , because many e rror sources are involved.

Th e flight test method could be particularly useful for rapid - prototype aircraft , for efficient stability and control flight testing , or in situations whe re the aircraft design or configuration are changed often , or as a cross - check for ground - test or CAD results. The capability to accurately estimate moments of inertia from flight test data can speed up aircraft development and decrease cost and risk by elim inating the need for ground testing or CAD modeling to obtain accurate moments of inertia. Because the method can be implemented in real time, other possible applications include real - time moment of inertia estimates for stores separation o r changing fuel or loading conditions.

Acknowledgments This research in a ircraft s ystem i dentification wa s funded by the NASA Transformational Tools and Technologies (TTT) project. The efforts of the E1 flight test team at NASA Langley in building and testing the aircraft and associated systems , carefully calibrating the instrumentation, and carrying out the flight operations to collect the high - quality flight test data used in this study are gratefully acknowledged. Dan Murri was the research pilot who skillfully flew the flight test maneuvers on the E1 aircraft. Ron Busan did the detailed ground testing and c alculations to generate the ground - test values of the E1 moments of inertia used for comparison with flight test results.

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