Piecewise Polynomial Model of the Aerodynamic Coefficients of the Generic Transport Model and its Equations of Motion

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Piecewise Polynomial Model of the Aerodynamic Coefficients of the Generic Transport Model and its

Equations of Motion

Torbjørn Cunis, Laurent Burlion, Jean-Philippe Condomines

To cite this version:

Torbjørn Cunis, Laurent Burlion, Jean-Philippe Condomines. Piecewise Polynomial Model of the

Aerodynamic Coefficients of the Generic Transport Model and its Equations of Motion. [Technical

Report] Third corrected version, ONERA – The French Aerospace Lab; École Nationale de l’Aviation

Civile. 2018. �hal-01808649v3�

Piecewise Polynomial Model of the Aerodynamic Coefficients of the Generic Transport Model and its Equations of Motion

Torbjørn Cunis

¹

, Laurent Burlion

²

, and Jean-Philippe Condomines

³

Abstract

The purpose of this document is to illustrate the piecewise polynomial model which has been derived from wind- tunnel measurement data of the Generic Transport Model (GTM) using the

pwpfit

toolbox. For implementation details and use in MATLAB, please refer to the source code at

https://github.com/pwpfit/GTMpw.

Preliminaries

If not stated otherwise, all variables are in SI units. We will refer to the following axis systems of ISO 1151-1:

the body axis system (x

_f

, y

_f

, z

_f

) aligned with the aircraft’s fuselage; the air-path axis system (x

_a

, y

_a

, z

_a

) defined by the velocity vector V

_A

; and the normal earth-fixed axis system (x

_g

, y

_g

, z

_g

) . The orientation of the body axes with respect to the normal earth-fixed system is given by the attitude angles Φ, Θ, Ψ and to the air-path system by angle of attack α and side-slip β ; the orientation of the air-path axes to the normal earth-fixed system is given by azimuth χ

_A

, inclination γ

_A

, and bank-angle µ

_A

. (Fig. 1.)

x

_g

x

_a

x

_f

z

_g

z

_a

z

_f

mg

−Z

^A

F

−X

^A

V

_A

Θ

γ

_A

α

(a) Longitudinal axes (

β=µ_A= 0

).

x

_g

x

⁰_f

x

_a

y

_g

y

_f⁰

y

_a

F

⁰

−X

^A

Y

^A0

V

_A

Ψ β

⁰

χ

_A

(b) Horizontal axes (

γ_A= 0

).

Fig. 1: Axis systems with angles and vectors. Projections into the plane are marked by

⁰

.

I. Aerodynamic Coefficients The piecewise polynomial models of the aerodynamic coefficients are given

C

(α, β, . . . ) =

C

^pre

(α, β, . . . ) if α ≤ α

0

,

C

^post

(α, β, . . . ) else, (1)

where C

∈ {C

_X

, C

_Y

, C

_Z

, C

_l

, C

_m

, C

_n

} are polynomials in angle of attack, side-slip, surface deflections, and normalized body rates; and the boundary is found at

α

0

= 16.111°. (2)

The polynomials in low and high angle of attack, C

^pre

, C

^post

, are sums

C

^pre

= C

_α^pre

(α) + C

_β^pre

(α, β) + C

_ξ^pre

(α, β, ξ) + C

_η^pre

(α, β, η) + C

_ζ^pre

(α, β, ζ) + C

_p^pre

(α, p) + ˆ C

_q^pre

(α, q) + ˆ C

_r^pre

(α, r) ; ˆ (3) C

^post

=C

_α^post

(α) +C

_β^post

(α, β) +C

_ξ^post

(α, β, ξ) +C

_η^post

(α, β, η) +C

_ζ^post

(α, β, ζ) +C

_p^post

(α, p) +C ˆ

_q^post

(α, q) +C ˆ

_r^post

(α, r) ˆ . (4)

1ONERA – The French Aerospace Lab, Department of Information Processing and Systems, Centre Midi-Pyrénées, Toulouse, 31055, Franceand ENAC, Université de Toulouse, Drones Research Group; now with the University of Michigan, e-mail:tcunis@umich.edu.

2Rutgers, The State University of New Jersey, Department of Mechanical & Aerospace Engineering, Piscataway, NJ 08854, USA, e-mail:laurent.burlion@rutgers.edu.

3ENAC, Université de Toulouse, Drones Research Group, Toulouse, 31055, France, e-mail:jean-philippe.condomines@enac.fr.

In the following subsections, we present the polynomial terms obtained using the pwpfit toolbox. Coefficients of absolute value lower than 10

⁻²

have been omitted for readability.

A. Domain of low angle of attack Polynomials in angle of attack:

C

_Xα^pre

= − 0 . 039 + 0 . 244 α + 4 . 452 α

²

− 17 . 394 α

³

; (5)

C

_Zα^pre

= −0.017 − 5.241α − 1.866α

²

+ 28.466α

³

; (6)

C

_m^pre_α

= 0 . 119 − 1 . 465 α + 8 . 129 α

²

− 31 . 983 α

³

; (7)

C

_Yα^pre

, C

_lα^pre

, C

_nα^pre

are zero by definition.

Polynomials in angle of attack and side-slip:

C

_Xβ^pre

=

0.012+0.013α−2.049α²+0.027αβ+0.066β²+9.808α³+0.249α²β−0.572αβ²−10.651α⁴−1.178α³β+1.936α²β²−0.043β⁴;

(8) C

_Yβ^pre

= − 1 . 024 β + 0 . 219 αβ − 0 . 307 α

²

β + 0 . 070 β

³

+ 4 . 667 α

³

β − 1 . 062 αβ

³

; (9) C

_Zβ^pre

=

− 0 . 035 − 0 . 018 α + 2 . 911 α

²

+ 0 . 024 β

²

− 6 . 328 α

³

− 0 . 098 α

²

β + 4 . 018 αβ

²

− 8 . 805 α

⁴

+ 0 . 328 α

³

β − 8 . 609 α

²

β

²

+ 0 . 165 β

⁴

; (10) C

_lβ^pre

= − 0 . 144 β + 0 . 177 αβ + 0 . 052 α

²

β + 0 . 203 β

³

+ 3 . 495 α

³

β − 1 . 113 αβ

³

; (11) C

_m^pre_β

=

0.068−0.205α−9.113α²+0.021αβ−1.396β²+54.917α³+0.170α²β−1.633αβ²−83.622α⁴−0.990α³β+11.760α²β²−0.027αβ³+1.164β⁴;

(12) C

_n^pre_β

= 0 . 228 β − 0 . 228 αβ − 0 . 230 β

³

− 1 . 697 α

³

β + 0 . 539 αβ

³

. (13) Polynomials in angle of attack, side-slip, and aileron deflections:

C

_Xξ^pre

=

0.033α−0.262α²+0.042β²+0.014βξ+0.135ξ²+1.195α³+0.094α²β−0.147αβ²+0.016αβξ−0.129αξ²−2.732α⁴−0.308α³β+0.993α²β²−0.575α²βξ+0.161α²ξ²−0.111β⁴−0.048β²ξ²−0.012βξ³−0.499ξ⁴;

(14) C

_Yξ^pre

=

0.080β−0.022ξ−1.925αβ−0.036αξ+0.501α²β+0.229α²ξ−0.021β³−0.244βξ²−1.469α³β−0.751α³ξ+3.189αβ³+5.135αβξ²+0.227αξ³;

(15) C

_Zξ^pre

=

−0.030α−0.322α²−0.033β²−0.140βξ+0.029ξ²+3.097α³+0.094α²β−0.328αβ²+0.385αβξ−0.247αξ²−5.289α⁴−0.308α³β+0.142α²β²+0.169α²βξ−0.795α²ξ²+0.109β⁴+0.027β³ξ−0.058β²ξ²+0.197βξ³+0.118ξ⁴;

(16) C

_lξ^pre

=

0.021β−0.079ξ−0.158αβ+0.046αξ+0.149α²β+0.698α²ξ−0.066β³+0.027β²ξ−0.036βξ²+0.069ξ³−0.508α³β−1.316α³ξ+0.338αβ³+0.016αβ²ξ+0.453αβξ²−0.139αξ³;

(17) C

_mξ^pre

=

−0.039+0.059α+0.163α²+0.401β²−0.013βξ−0.564ξ²+2.568α³+0.094α²β−1.548αβ²+0.068αβξ−0.439αξ²−4.594α⁴−0.308α³β−1.295α²β²−0.048α²βξ+0.356α²ξ²+0.085β⁴+0.082β³ξ+0.084β²ξ²+0.038βξ³+2.201ξ⁴;

(18) C

_nξ^pre

=

−0.249αβ+0.014αξ+0.041α²β+0.053α²ξ+0.029β³−0.017βξ²+0.016ξ³+0.337α³β+0.048α³ξ+0.337αβ³+0.049αβ²ξ+0.690αβξ²−0.122αξ³.

(19)

Polynomials in angle of attack, side-slip, and elevator deflections:

C

_Xη^pre

= −0 . 036 α − 0 . 292 α

²

+ 0 . 148 αη − 0 . 102 η

²

+ 1 . 173 α

³

− 0 . 411 α

²

η + 0 . 033 αβ

²

− 0 . 011 αβη + 0 . 105 αη

²

− 0 . 036 η

³

; (20) C

_Y^pre_η

= − 0 . 267 β + 1 . 036 αβ − 3 . 141 α

²

β + 0 . 532 β

³

; (21) C

_Zη^pre

=

0.041α−0.016β−0.526η+0.205α²+0.074αβ+0.050αη−0.012β²−1.770α³−0.060α²β+0.815α²η+0.306αβ²−0.028αβη+0.181αη²+0.297β²η+0.012βη²+0.645η³;

(22) C

_lη^pre

= −0.012β + 0.057αβ − 0.286α

²

β + 0.047β

³

; (23) C

_mη^pre

=

0.028+0.032α−0.054β−1.851η−0.172α²+0.206αβ−0.172αη−0.199β²+0.026βη−0.173η²−0.300α³−0.068α²β+5.136α²η+0.816αβ²−0.023αβη+0.897αη²+0.693β²η+0.111βη²+1.324η³;

(24) C

_nη^pre

= −0.057β + 0.142αβ − 0.344α

²

β + 0.110β

³

. (25) Polynomials in angle of attack, side-slip, and rudder deflections:

C

_X^pre_ζ

=

−0.011+0.026α+0.351α²+0.022β²+0.091βζ−2.731α³−0.042αβ²−0.040αβζ+0.013αζ²+6.059α⁴+0.053α²β²+0.245α²βζ−0.211α²ζ²−0.011β⁴−0.042β²ζ²−0.130βζ³−0.056ζ⁴;

(26) C

_Y^pre_ζ

=

0.187β+0.308ζ+0.820αβ−0.501αζ+0.160α²β+1.251α²ζ−0.455β³−0.224β²ζ+0.168βζ²−0.235ζ³+14.007α³β−6.802α³ζ−2.757αβ³+0.307αβ²ζ−0.678αβζ²+1.093αζ³;

(27) C

_Zζ^pre

=

−0.106α+0.016β+0.254α²−0.074αβ+0.055β²+0.054βζ+0.117ζ²+3.459α³+0.060α²β+0.089αβ²−0.367αβζ+0.063αζ²−6.397α⁴−0.984α²β²+1.934α²βζ−1.049α²ζ²−0.099β⁴−0.140β³ζ−0.047β²ζ²−0.043βζ³−0.122ζ⁴;

(28) C

_lζ^pre

=

0.024ζ+0.027αβ−0.027αζ+0.150α²β+0.068α²ζ−0.013β³−0.026β²ζ−0.017ζ³+0.600α³β−0.251α³ζ−0.172αβ³+0.046αβ²ζ+0.020αβζ²+0.040αζ³;

(29) C

_mζ^pre

=

−0.016−0.039α+0.054β+0.335α²−0.206αβ+0.032β²−0.340βζ+0.298ζ²−7.653α³+0.068α²β+0.723αβ²−0.354αβζ+0.021αζ²+23.281α⁴−1.037α²β²+0.588α²βζ−1.013α²ζ²−0.311β⁴−0.304β³ζ+0.213β²ζ²+0.680βζ³−0.247ζ⁴;

(30) C

_nζ^pre

=

0.040β−0.145ζ+0.109αβ+0.033αζ+0.606α²β−0.264α²ζ−0.108β³+0.114β²ζ−0.052βζ²+0.088ζ³−1.391α³β+1.698α³ζ−0.387αβ³−0.098αβ²ζ+0.077αβζ²−0.090αζ³.

(31) Polynomials in angle of attack and normalized body p-rate:

C

_Y^pre_pˆ

= −0.071 p ˆ − 0.138αˆ p − 0.013ˆ p

²

− 0.862α

²

p ˆ + 0.383αˆ p

²

+ 14.142 p ˆ

³

; (32) C

_lˆ^pre_p

= −0.266 p ˆ − 0.247αˆ p − 0.015ˆ p

²

+ 3.159α

²

p ˆ + 0.540αˆ p

²

− 2.843 p ˆ

³

; (33) C

_nˆ^pre_p

= − 0 . 083 p ˆ − 0 . 044 αˆ p − 0 . 065 p ˆ

²

+ 1 . 221 α

²

p ˆ + 0 . 073 αˆ p

²

+ 5 . 811 p ˆ

³

; (34) C

_Xˆ^pre_p

, C

_Zˆ^pre_p

, C

_mˆ^pre_p

are zero due to lack of GTM measurement data.

Polynomials in angle of attack and normalized body q -rate:

C

_Xˆ^pre_q

= − 0 . 030 α + 0 . 851 q ˆ + 0 . 076 α

²

+ 12 . 434 αˆ q + 571 . 330 q ˆ

²

+ 0 . 127 α

³

+ 23 . 482 α

²

q ˆ + 2196 . 700 αˆ q

²

+ 2529 . 200 q ˆ

³

; (35) C

_Zˆ^pre_q

= −32.946ˆ q − 0.203α

²

− 32.151αˆ q + 1401.800ˆ q

²

− 0.282α

³

− 80.578α

²

q ˆ + 1239.200αˆ q

²

+ 2345.900ˆ q

³

; (36) C

_mˆ^pre_q

=

− 0 . 024 − 0 . 033 α − 42 . 378 q ˆ + 0 . 496 α

²

− 2 . 991 αˆ q + 765 . 020 q ˆ

²

+ 0 . 909 α

³

+ 29 . 585 α

²

q ˆ + 2214 . 800 αˆ q

²

+ 2370 . 400 q ˆ

³

;

(37)

C

_Yˆ^pre_q

, C

_lˆ^pre_q

, C

_nˆ^pre_q

are zero due to lack of GTM measurement data.

Polynomials in angle of attack and normalized body r -rate:

C

_Yˆ^pre_r

= 0 . 748 r ˆ + 0 . 865 αˆ r + 0 . 881 ˆ r

²

+ 1 . 625 α

²

r ˆ − 1 . 485 αˆ r

²

+ 2 . 662 r ˆ

³

; (38) C

_l^pre_rˆ

= 0 . 147 r ˆ + 0 . 755 αˆ r + 0 . 862 α

²

r ˆ + 0 . 125 αˆ r

²

− 5 . 986 r ˆ

³

; (39) C

_nˆ^pre_r

= −0 . 337 ˆ r − 0 . 249 αˆ r − 0 . 471 r ˆ

²

− 0 . 912 α

²

ˆ r + 1 . 028 αˆ r

²

+ 19 . 902 r ˆ

³

; (40) C

_Xˆ^pre_r

, C

_Zˆ^pre_r

, C

_mˆ^pre_r

are zero due to lack of GTM measurement data.

B. Domain of high angle of attack Polynomials in angle of attack:

C

_X^post_α

= 0 . 019 − 0 . 130 α + 0 . 169 α

²

− 0 . 022 α

³

; (41)

C

_Zα^post

= −0.365 − 2.712α + 1.647α

²

− 0.369α

³

; (42)

C

_mα^post

= 0 . 247 − 2 . 847 α + 2 . 748 α

²

− 1 . 105 α

³

; (43)

C

_Y^post_α

, C

_lα^post

, C

_nα^post

are zero by definition.

Polynomials in angle of attack and side-slip:

C

_Xβ^post

= 0 . 101 α − 0 . 239 α

²

+ 0 . 025 β

²

+ 0 . 199 α

³

+ 0 . 160 αβ

²

− 0 . 011 β

³

− 0 . 055 α

⁴

− 0 . 148 α

²

β

²

+ 0 . 010 αβ

³

− 0 . 043 β

⁴

; (44) C

_Yβ^post

= −0.393β − 2.181αβ + 1.680α

²

β − 0.591β

³

− 0.402α

³

β + 1.287αβ

³

; (45) C

_Zβ^post

=

−0.040+0.183α−0.268α²+0.039αβ−0.293β²+0.168α³−0.088α²β+3.329αβ²−0.039α⁴+0.056α³β−2.148α²β²+0.165β⁴;

(46) C

_lβ^post

= 0 . 074 β − 0 . 339 αβ + 0 . 111 α

²

β − 0 . 142 β

³

+ 0 . 113 αβ

³

; (47) C

_mβ^post

=

0.357−2.954α−0.096β+7.802α²+0.510αβ−0.589β²−7.667α³−0.736α²β−1.740αβ²+2.469α⁴+0.296α³β+1.931α²β²−0.040αβ³+1.164β⁴;

(48) C

_nβ^post

= 0 . 197 β − 0 . 186 αβ − 0 . 286 α

²

β − 0 . 238 β

³

+ 0 . 160 α

³

β + 0 . 567 αβ

³

. (49) Polynomials in angle of attack, side-slip, and aileron deflections:

C

_Xξ^post

=

0.079−0.611α+1.476α²+0.135β²−0.025βξ−0.083ξ²−1.489α³−0.253αβ²−0.013αβξ+0.897αξ²+0.531α⁴+0.194α²β²+0.026α²βξ−0.717α²ξ²−0.111β⁴−0.048β²ξ²−0.012βξ³−0.499ξ⁴;

(50) C

_Yξ^post

=

−1.486β−0.059ξ+4.744αβ+0.127αξ−4.548α²β−0.097α²ξ+2.041β³+2.727βξ²+0.121ξ³+2.605α³β+0.031α³ξ−4.143αβ³+0.013αβ²ξ−5.429αβξ²−0.185αξ³;

(51) C

_Zξ^post

=

0.038−0.332α+0.762α²−0.043β²−0.277ξ²−0.480α³−0.326αβ²−0.034αβξ+0.676αξ²+0.054α⁴+0.262α²β²−0.033α²βξ−0.206α²ξ²+0.109β⁴+0.027β³ξ−0.058β²ξ²+0.197βξ³+0.118ξ⁴;

(52) C

_lξ^post

=

−0.107β−0.035ξ+0.419αβ−0.042αξ−0.500α²β+0.084α²ξ+0.082β³+0.035β²ξ+0.194βξ²+0.030ξ³+0.253α³β−0.032α³ξ−0.188αβ³−0.013αβ²ξ−0.365αβξ²;

(53) C

_m^post_ξ

=

−0.062+0.636α−1.619α²−0.520β²+0.051βξ−0.061ξ²+1.436α³+1.727αβ²−0.202αβξ−2.612αξ²−0.434α⁴−1.297α²β²+0.105α²βξ+1.727α²ξ²+0.085β⁴+0.082β³ξ+0.084β²ξ²+0.038βξ³+2.201ξ⁴;

(54) C

_n^post_ξ

=

−0.170β+0.013ξ+0.458αβ−0.034αξ−0.227α²β+0.037α²ξ+0.328β³+0.017β²ξ+0.397βξ²−0.045ξ³+0.161α³β−0.021α³ξ−0.726αβ³−0.017αβ²ξ−0.781αβξ²+0.095αξ³.

(55)

Polynomials in angle of attack, side-slip, and elevator deflections:

C

_Xη^post

=

0.042 − 0.276α − 0.011η + 0.546α

²

+ 0.061αη + 0.039β

²

− 0.063η

²

− 0.293α

³

− 0.063α

²

η − 0.088αβ

²

− 0.032αη

²

− 0.036η

³

; (56) C

_Yη^post

= − 0 . 608 β + 1 . 973 αβ − 2 . 152 α

²

β + 0 . 532 β

³

; (57) C

_Z^post_η

=

−0.100+0.559α−0.594η−0.926α²+0.029αβ+0.585αη+0.110β²−0.035η²+0.452α³−0.024α²β−0.230α²η−0.127αβ²+0.319αη²+0.297β²η+0.012βη²+0.645η³;

(58) C

_l^post_η

= − 0 . 049 β + 0 . 150 αβ − 0 . 152 α

²

β + 0 . 047 β

³

; (59) C

_m^post_η

=

0.090−0.372α−1.880η+0.436α²+1.470αη+0.019βη−0.087η²−0.139α³−0.341α²η+0.076αβ²+0.589αη²+0.693β²η+0.111βη²+1.324η³;

(60) C

_nη^post

= − 0 . 070 β + 0 . 142 αβ − 0 . 187 α

²

β + 0 . 110 β

³

. (61) Polynomials in angle of attack, side-slip, and rudder deflections:

C

_Xζ^post

=

0.016−0.140α+0.478α²+0.082β²+0.203βζ−0.122ζ²−0.669α³−0.316αβ²−0.463αβζ+0.456αζ²+0.298α⁴+0.274α²β²+0.325α²βζ−0.307α²ζ²−0.011β⁴−0.042β²ζ²−0.130βζ³−0.056ζ⁴;

(62) C

_Yζ^post

=

3.051β−0.205ζ−11.790αβ+1.702αζ+14.450α²β−2.320α²ζ−2.302β³−0.260β²ζ−0.182βζ²+0.353ζ³−6.130α³β+1.104α³ζ+3.811αβ³+0.433αβ²ζ+0.566αβζ²−0.996αζ³;

(63) C

_Zζ^post

=

0.220−1.248α+2.437α²−0.029αβ−0.083β²+0.142βζ−0.081ζ²−2.074α³+0.024α²β+0.345αβ²−0.163αβζ+0.591αζ²+0.654α⁴−0.149α²β²+0.107α²βζ−0.422α²ζ²−0.099β⁴−0.140β³ζ−0.047β²ζ²−0.043βζ³−0.122ζ⁴;

(64) C

_lζ^post

=

0.129β−0.471αβ+0.047αζ+0.546α²β−0.070α²ζ−0.091β³−0.019β²ζ+0.012βζ²−0.203α³β+0.031α³ζ+0.106αβ³+0.021αβ²ζ−0.010αβζ²−0.033αζ³;

(65) C

_mζ^post

=

0.429−3.325α+8.084α²+0.245β²−0.974βζ+0.199ζ²−7.822α³−0.363αβ²+2.617αβζ+0.150αζ²+2.615α⁴+0.142α²β²−1.957α²βζ−0.220α²ζ²−0.311β⁴−0.304β³ζ+0.213β²ζ²+0.680βζ³−0.247ζ⁴;

(66) C

_nζ^post

=

0.149β−0.146ζ−0.223αβ+0.055αζ+0.015α²β+0.184α²ζ−0.377β³+0.176β²ζ−0.036βζ²+0.067ζ³−0.013α³β−0.106α³ζ+0.568αβ³−0.318αβ²ζ+0.020αβζ²−0.015αζ³.

(67) Polynomials in angle of attack and normalized body p -rate:

C

_Yˆ^post_p

= − 0 . 058 p ˆ − 0 . 590 αˆ p + 0 . 970 p ˆ

²

+ 0 . 582 α

²

p ˆ − 3 . 112 αˆ p

²

+ 14 . 142 p ˆ

³

; (68)

C

_l^post_pˆ

= 0.044 p ˆ − 0.589αˆ p + 0.133ˆ p

²

+ 0.464α

²

p ˆ + 0.013αˆ p

²

− 2.843 p ˆ

³

; (69)

C

_nˆ^post_p

= 0.113 p ˆ − 0.504αˆ p − 0.240ˆ p

²

+ 0.369α

²

p ˆ + 0.695αˆ p

²

+ 5.811 p ˆ

³

; (70)

C

_Xˆ^post_p

, C

_Zˆ^post_p

, C

_mˆ^post_p

are zero due to lack of GTM measurement data.

Polynomials in angle of attack and normalized body q -rate:

C

_Xˆ^post_q

=

0 . 034 − 0 . 209 α + 22 . 961 q ˆ + 0 . 246 α

²

− 76 . 364 αˆ q + 821 . 140 q ˆ

²

− 0 . 097 α

³

+ 59 . 647 α

²

q ˆ + 1308 . 300 αˆ q

²

+ 2529 . 200 q ˆ

³

; (71) C

_Zˆ^post_q

=

0.015 − 0.385α − 91.233ˆ q + 0.978α

²

+ 199.050αˆ q + 1312.200ˆ q

²

− 0.677α

³

− 165.610α

²

q ˆ + 1557.900αˆ q

²

+ 2345.900 q ˆ

³

; (72) C

_mˆ^post_q

=

0.128 − 0.470α − 11.370ˆ q + 0.432α

²

− 145.970αˆ q + 1029.300ˆ q

²

− 0.173α

³

+ 145.910α

²

q ˆ + 1275.000αˆ q

²

+ 2370.400 q ˆ

³

; (73) C

_Yˆ^post_q

, C

_lˆ^post_q

, C

_nˆ^post_q

are zero due to lack of GTM measurement data.

Polynomials in angle of attack and normalized body r-rate:

C

_Yˆ^post_r

= 3 . 868 ˆ r − 11 . 818 αˆ r + 1 . 677 ˆ r

²

+ 7 . 267 α

²

r ˆ − 4 . 315 αˆ r

²

+ 2 . 662 r ˆ

³

; (74) C

_lˆ^post_r

= − 0 . 154 r ˆ + 3 . 156 αˆ r + 0 . 019 r ˆ

²

− 3 . 867 α

²

r ˆ + 0 . 042 αˆ r

²

− 5 . 986 ˆ r

³

; (75) C

_nˆ^post_r

= −0.639ˆ r + 0.714αˆ r − 0.239 r ˆ

²

− 0.510α

²

r ˆ + 0.203αˆ r

²

+ 19.902ˆ r

³

; (76) C

_Xˆ^post_r

, C

_Zˆ^post_r

, C

_mˆ^post_r

are zero due to lack of GTM measurement data.

II. Equations of Motion We have the aerodynamic forces and moments in body axis system



 X

^A

Y

^A

Z

^A





f

= 1 2 %SV

_A²



 C

_X

C

_Y

C

_Z



 ;



 L

^A

M

^A

N

^A





f

= 1 2 %SV

_A²



 b C

_l

c

_a

C

_m

b C

_n



 +



 X

^A

Y

^A

Z

^A





f

× x

_cg

− x

^ref_cg

; (77)

and the weight force by rotation into the body axis system

X

_g^G

= −g sin Θ; (78)

Y

_g^G

= −g sin Φ cos Θ; (79)

Z

_g^G

= −g cos Φ cos Θ; (80)

The resulting forces lead to changes in the velocity vector, in body axis system, by

V ˙

_Af

= 1 m





X

^A

+ X

^G

+ X

^F

Y

^A

+ Y

^G

Z

^A

+ Z

^G





f

+



 p q r



 × V

_Af

. (81) with the thrust X

_f^F

= F (engines aligned with the x

_f

-axis). For a symmetric plane (I

xy

= I

yz

= 0), the resulting moments in body axis are given as

L

_f

= L

^A_f

− q r (I

_z

− I

_y

) + p q I

_zx

; (82)

M

_f

= M

_f^A

+ M

_f^F

− p r (I

x

− I

z

) − p

²

− r

²

I

zx

; (83)

N

_f

= N

_f^A

− p q (I

_y

− I

_x

) − q r I

_zx

; (84)

with M

_f^F

= l

_t

F (engines symmetric to the x

_f

- y

_f

-plane and shifted vertically from the origin by l

_t

) and the inertias I

x

, I

y

, I

z

, I

zx

. The changes of angular body rates are then given as

˙

p = 1

I

x

I

z

− I

_zx²

(I

z

L

_f

+ I

zx

N

_f

) ; (85)

˙ q = 1

I

_y

M

_f

; (86)

˙

r = 1

I

x

I

z

− I

_zx²

(I

zx

L

_f

+ I

x

N

_f

) . (87) Here, the normalized body rates have been used with



 ˆ p ˆ q ˆ r



 = 1 2V

_A



 b p c

_a

q

b r



 ⇐⇒



 p q r



 = 2V

_A



 b

⁻¹

p ˆ c

⁻¹_a

q ˆ b

⁻¹

ˆ r



 (88)

and





˙ˆ p

˙ˆ q

˙ˆ r



 = 1 V

_A



 1 2



 b p ˙ c

_a

q ˙ b r ˙



 − V ˙

_A



 ˆ p ˆ q ˆ r







 . (89)

The change of attitude is finally obtained by rotation into normal earth-fixed axis system:

Φ = ˙ p + q sin Φ tan Θ + r cos Φ tan Θ; (90)

Θ = ˙ q cos Φ − r sin Φ; (91)

Ψ = ˙ q sin Φ cos

⁻¹

Θ + r cos Φ cos

⁻¹

Θ. (92) III. Longitudinal Model

The longitudinal model is restricted to the x

_a

- z

_a

-plane, assuming β = µ

_A

= χ

_A

= 0 . A. Aerodynamic Coefficients

The longitudinal aerodynamic coefficients, C

_L

, C

_D

, C

_m

, are obtained as C

_L

(α, η) = C

_Lα

(α) + C

_Lη

(α, η) with

C

_Lα

(α) =

0 . 017 + 5 . 234 α + 1 . 985 α

²

− 30 . 060 α

³

if α ≤ 16 . 634°

0 . 279 + 3 . 251 α − 3 . 235 α

²

+ 0 . 708 α

³

else , (93) C

_Lη

(α, η) = − 0 . 000 + 0 . 003 α + 0 . 521 η − 0 . 072 α

²

− 0 . 416 αη

+ 0 . 089 η

²

+ 0 . 051 α

³

+ 0 . 039 α

²

η − 0 . 293 αη

²

− 0 . 479 η

³

; (94) C

_D

(α, η) = C

_Dα

(α) + C

_Dη

(α, η) with

C

_Dα

(α) =

0.029 − 0.110α + 2.364α

²

+ 3.948α

³

if α ≤ 16.634°

−0.170 + 1.427α + 0.719α

²

− 0.486α

³

else , (95) C

_Dη

(α, η) = + 0 . 008 − 0 . 012 α + 0 . 112 η + 0 . 040 α

²

+ 0 . 183 αη

− 0 . 069 η

²

− 0 . 053 α

³

− 0 . 043 α

²

η − 0 . 070 αη

²

− 0 . 628 η

³

; (96) C

_m

(α, η) = C

_mα

(α) + C

_mη

(α, η) with

C

_mα

(α) =

0 . 117 − 1 . 475 α + 8 . 475 α

²

− 32 . 729 α

³

if α ≤ 16 . 634°

0 . 144 − 2 . 456 α + 2 . 304 α

²

− 0 . 950 α

³

else , (97) C

_mη

(α, η) = + 0.014 + 0.165α − 1.968η − 0.410α

²

+ 1.365αη

− 0 . 415 η

²

+ 0 . 186 α

³

− 0 . 144 α

²

η + 0 . 948 αη

²

+ 1 . 356 η

³

. (98)

B. Equations of motion

The longitudinal equations of motion are given as V ˙

_A

= 1

m

F cos α − 1

2 ρSV

_A²

C

_D

(α, η, q) − mg sin γ

_A

, (99)

˙

γ

_A

= 1 mV

_A

F sin α + 1

2 ρSV

_A²

C

_L

(α, η, q) − mg cos γ

_A

, (100)

˙ q = 1

I

_y

l

_t

F + 1

2 ρSc

_a

V

_A²

C

_m

(α, η, q) − 1

2 ρSV

_A²

C

_Z

(α, η, q) x

^ref_cg

− x

_cg

+ 1

2 ρSV

_A²

C

_X

(α, η, q) z

_cg^ref

− z

_cg

; (101) with

Θ = α + γ. (102)

MATLAB Source Code

The source code for the aerodynamic coefficients and the equations of motion can be found at:

https://github.com/pwpfit/GTMpw

Appendix A. Spline-based Longitudinal Coefficients

The purpose of this model is to provide an application example of a spline-based longitudinal aircraft model. It has neither been designed nor evaluated for engineering purposes. For the longitudinal equations of motion, refer to Section III-B and note

C

_D

(α, . . .) = −C

_X

(α, . . .) cos α − C

_Z

(α, . . .) sin α (103) C

_L

(α, . . .) = C

_X

(α, . . .) sin α − C

_Z

(α, . . .) cos α (104) The longitudinal aerodynamic coefficients are obtained as

C

(α, η, q) = ˆ C

_α

(α) + C

_η

(α, η) + C

_qˆ

(α, q) ˆ , (105) where C ∈ {C

_X

, C

_Z

, C

_m

}, q ˆ = c

_a

q/ (2V

_A

), and

C

α

(α, η) =



 

 

 

 

 

 

C

_α⁽¹⁾

if α ∈ (−∞; α

1

) , C

_α⁽²⁾

if α ∈ [α

1

; α

2

) , C

_α⁽³⁾

if α ∈ [α

2

; α

3

) , C

_α⁽⁴⁾

if α ∈ [α

3

; α

3

) , C

_α⁽⁵⁾

if α ∈ [α

4

; ∞);

(106)

C

_η

(α, η) =



 

 

 



 

 

 



C

_η⁽¹⁾

if α ∈ (−∞; α

₁

) , C

_η⁽²⁾

if α ∈ [α

1

; α

2

), C

_η⁽³⁾

if α ∈ [α

₂

; α

₃

) , C

_η⁽⁴⁾

if α ∈ [α

3

; α

3

), C

_η⁽⁵⁾

if α ∈ [α

4

; ∞);

(107)

C

_ˆq

(α, q) = ˆ



 

 

 



 

 

 



C

_ˆ⁽¹⁾_q

if q ˆ ∈ (−∞; ˆ q

1

) , C

_ˆ⁽²⁾_q

if q ˆ ∈ [ˆ q

1

; ˆ q

2

) , C

_ˆ⁽³⁾_q

if q ˆ ∈ [ˆ q

2

; ˆ q

3

) , C

_ˆ⁽⁴⁾_q

if q ˆ ∈ [ˆ q

₃

; ˆ q

₃

) , C

_ˆ⁽⁵⁾_q

if q ˆ ∈ [ˆ q

₄

; ∞);

(108)

with

α

₁

= 5° , α

₂

= 15° , α

₃

= 25° , α

₄

= 45° ; (109)

ˆ

q

1

= − 0 . 200° , q ˆ

2

= − 0 . 075° , q ˆ

3

= 0 . 075° , q ˆ

4

= 0 . 200° ; (110) and

C

_X⁽¹⁾_α

(α) = − 0 . 025 − 0 . 002 α + 0 . 827 α

²

; (111)

C

_Xα⁽²⁾

(α) = −0 . 204 + 2 . 778 α − 7 . 493 α

²

; (112)

C

_X⁽³⁾_α

(α) = 0 . 217 − 1 . 218 α + 1 . 628 α

²

; (113)

C

_Xα⁽⁴⁾

(α) = 0 . 022 − 0 . 110 α + 0 . 114 α

²

; (114)

C

_X⁽⁵⁾_α

(α) = −0.052 + 0.024α + 0.063α

²

; (115)

C

_Z⁽¹⁾_α

(α) = − 0 . 028 − 4 . 949 α + 0 . 837 α

²

; (116)

C

_Zα⁽²⁾

(α) = 0 . 155 − 8 . 239 α + 14 . 537 α

²

; (117)

C

_Z⁽³⁾_α

(α) = − 0 . 815 − 0 . 298 α − 1 . 648 α

²

; (118)

C

_Zα⁽⁴⁾

(α) = −0 . 467 − 2 . 158 α + 0 . 786 α

²

; (119)

C

_Z⁽⁵⁾_α

(α) = − 1 . 052 − 0 . 996 α + 0 . 254 α

²

; (120)

C

_mα⁽¹⁾

(α) = 0 . 157 − 1 . 724 α + 1 . 806 α

²

+ 11 . 969 α

³

; (121) C

_mα⁽²⁾

(α) = 0 . 565 − 10 . 419 α + 59 . 277 α

²

− 118 . 320 α

³

; (122) C

_mα⁽³⁾

(α) = 7 . 543 − 64 . 304 α + 174 . 300 α

²

− 160 . 370 α

³

; (123) C

_mα⁽⁴⁾

(α) = − 0 . 866 + 1 . 176 α − 1 . 875 α

²

+ 0 . 681 α

³

; (124) C

_mα⁽⁵⁾

(α) = −1.053 + 1.752α − 2.210α

²

+ 0.561α

³

; (125) C

_X⁽¹⁾_η

(α, η) = − 0 . 010 η + 0 . 195 αη − 0 . 082 η

²

+ 0 . 628 α

²

η + 0 . 213 αη

²

− 0 . 036 η

³

; (126) C

_Xη⁽²⁾

(α, η) = 0 . 011 η − 0 . 009 αη − 0 . 060 η

²

+ 0 . 165 α

²

η − 0 . 043 αη

²

− 0 . 036 η

³

; (127) C

_X⁽³⁾_η

(α, η) = 0 . 060 η − 0 . 175 αη − 0 . 043 η

²

+ 0 . 087 α

²

η − 0 . 106 αη

²

− 0 . 036 η

³

; (128) C

_Xη⁽⁴⁾

(α, η) = 0 . 027 η − 0 . 079 αη − 0 . 056 η

²

+ 0 . 042 α

²

η − 0 . 076 αη

²

− 0 . 036 η

³

; (129) C

_Xη⁽⁵⁾

(α, η) = − 0 . 281 η + 0 . 586 αη − 0 . 133 η

²

− 0 . 305 α

²

η + 0 . 021 αη

²

− 0 . 036 η

³

; (130) C

_Zη⁽¹⁾

(α, η) = − 0 . 595 η + 0 . 058 αη + 0 . 039 η

²

+ 0 . 045 α

²

η + 0 . 023 αη

²

+ 1 . 142 η

³

; (131) C

_Zη⁽²⁾

(α, η) = −0.626η + 0.461αη + 0.012η

²

− 0.452α

²

η + 0.332αη

²

+ 1.142η

³

; (132) C

_Zη⁽³⁾

(α, η) = − 0 . 631 η + 0 . 299 αη + 0 . 051 η

²

+ 0 . 224 α

²

η + 0 . 183 αη

²

+ 1 . 142 η

³

; (133) C

_Z⁽⁴⁾_η

(α, η) = −0.666η + 0.516αη − 0.337η

²

− 0.084α

²

η + 1.074αη

²

+ 1.142η

³

; (134) C

_Zη⁽⁵⁾

(α, η) = − 1 . 116 η + 1 . 625 αη + 0 . 312 η

²

− 0 . 768 α

²

η + 0 . 247 αη

²

+ 1 . 142 η

³

; (135) C

_mη⁽¹⁾

(α, η) = − 1 . 879 η − 0 . 047 αη − 0 . 227 η

²

− 0 . 845 α

²

η + 0 . 378 αη

²

+ 1 . 409 η

³

; (136) C

_mη⁽²⁾

(α, η) = − 2 . 006 η + 1 . 410 αη − 0 . 260 η

²

− 0 . 807 α

²

η + 0 . 756 αη

²

+ 1 . 409 η

³

; (137) C

_mη⁽³⁾

(α, η) = − 1 . 916 η − 0 . 071 αη − 0 . 704 η

²

+ 3 . 530 α

²

η + 2 . 449 αη

²

+ 1 . 409 η

³

; (138) C

_m⁽⁴⁾_η

(α, η) = −1.612η + 0.765αη + 0.802η

²

+ 0.017α

²

η − 1.002αη

²

+ 1.409η

³

; (139) C

_mη⁽⁵⁾

(α, η) = −1.555η + 0.535αη − 0.777η

²

+ 0.219α

²

η + 1.009αη

²

+ 1.409η

³

; (140) as well as

C

_X⁽¹⁾_qˆ

(α, q) = ˆ − 0 . 003 − 1 . 776 q ˆ − 0 . 010 α − 276 . 480 q ˆ

⁽²⁾

− 2 . 319 qα ˆ − 0 . 005 α

²

; (141)

C

_Xˆ⁽²⁾_q

(α, q) = ˆ 0 . 001 + 0 . 689 q ˆ + 0 . 004 α + 120 . 080 q ˆ

⁽²⁾

+ 1 . 492 qα ˆ − 0 . 005 α

²

; (142)

C

_Xˆ⁽³⁾_q

(α, q) = ˆ 0 . 001 + 1 . 338 q ˆ + 0 . 003 α + 232 . 350 q ˆ

⁽²⁾

+ 0 . 558 qα ˆ − 0 . 005 α

²

; (143)

C

_X⁽⁴⁾_qˆ

(α, q) = ˆ 0.001 + 1.897ˆ q + 0.004α + 144.900 q ˆ

⁽²⁾

− 0.699ˆ qα − 0.005α

²

; (144)

C

_Xˆ⁽⁵⁾_q

(α, q) = ˆ 0 . 001 + 2 . 271 q ˆ − 0 . 009 α − 7 . 040 q ˆ

⁽²⁾

+ 2 . 981 qα ˆ − 0 . 005 α

²

; (145)

C

_Z⁽¹⁾_qˆ

(α, q) = ˆ −0.008 − 25.814ˆ q + 0.138α + 2806.400ˆ q

⁽²⁾

+ 22.164ˆ qα + 0.004α

²

; (146)

C

_Zˆ⁽²⁾_q

(α, q) = ˆ − 0 . 010 − 32 . 148 q ˆ + 0 . 019 α + 1120 . 800 q ˆ

⁽²⁾

− 11 . 921 qα ˆ + 0 . 004 α

²

; (147)

C

_Zˆ⁽³⁾_q

(α, q) = ˆ − 0 . 014 − 36 . 128 q ˆ + 0 . 029 α + 507 . 060 q ˆ

⁽²⁾

− 3 . 943 qα ˆ + 0 . 004 α

²

; (148)

C

_Zˆ⁽⁴⁾_q

(α, q) = ˆ −0.010 − 40.109ˆ q + 0.019α + 1120.800ˆ q

⁽²⁾

+ 4.036 qα ˆ + 0.004α

²

; (149)

C

_Zˆ⁽⁵⁾_q

(α, q) = ˆ − 0 . 008 − 46 . 443 q ˆ + 0 . 138 α + 2806 . 400 q ˆ

⁽²⁾

− 30 . 049 qα ˆ + 0 . 004 α

²

; (150)

C

_mˆ⁽¹⁾_q

(α, q) = ˆ 0.008 − 41.631 q ˆ − 0.013α + 300.350ˆ q

⁽²⁾

− 10.716ˆ qα − 0.081α

²

; (151)

C

_mˆ⁽²⁾_q

(α, q) = ˆ 0 . 009 − 40 . 268 q ˆ + 0 . 009 α + 632 . 510 q ˆ

⁽²⁾

− 4 . 284 qα ˆ − 0 . 081 α

²

; (152)

C

_m⁽³⁾_ˆq

(α, q) = ˆ 0.009 − 40.445 q ˆ + 0.012α + 674.850ˆ q

⁽²⁾

− 2.169 qα ˆ − 0.081α

²

; (153)

C

_mˆ⁽⁴⁾_q

(α, q) = ˆ 0 . 009 − 39 . 916 q ˆ + 0 . 009 α + 381 . 400 q ˆ

⁽²⁾

− 0 . 193 qα ˆ − 0 . 081 α

²

; (154)

C

_mˆ⁽⁵⁾_q

(α, q) = ˆ − 0 . 001 − 35 . 258 q ˆ + 0 . 008 α − 208 . 640 q ˆ

⁽²⁾

+ 0 . 192 qα ˆ − 0 . 081 α

²

; (155)

Nomenclature α = Angle of attack (rad);

α

0

= Low-angle of attack boundary (°);

β = Side-slip angle (rad);

γ

_A

= Air-path inclination angle (rad);

ζ = Rudder deflection (rad), negative if leading to positive yaw moment;

η = Elevator deflection (rad), negative if leading to positive pitch moment;

µ

_A

= Air-path bank angle (rad);

ξ = Aileron deflection (rad), negative if leading to positive roll moment;

% = Air density ( % = 1 . 200 kg m

⁻³

);

χ

A

= Air-path azimuth angle (rad);

Θ = Pitch angle (rad);

Φ = Bank angle (rad);

Ψ = Azimuth angle (rad);

b = Reference aerodynamic span ( b = 2 . 088 m);

c

_A

= Aerodynamic mean chord (c

A

= 0.280 m);

g = Standard gravitational acceleration (g ≈ 9.810 m s

⁻²

);

l

_t

= Engine vertical displacement, positive along z

_f

-axis (l

t

= 0.100 m);

m = Aircraft mass (m = 26.190 kg);

p = Roll rate (rad s

⁻¹

);

q = Pitch rate (rad s

⁻¹

);

r = Yaw rate (rad s

⁻¹

);

x

_cg

, z

_cg

= Longitudinal position center of gravity ( x

_cg

= − 1 . 450 m, z

_cg

= − 0 . 300 m);

x

^ref_cg

, z

^ref_cg

= Longitudinal position reference center of gravity ( x

^ref_cg

= − 1 . 460 m, z

_cg^ref

= − 0 . 290 m);

C

_l

= Aerodynamic coefficient moment body x

_f

-axis ( · );

C

_m

= Aerodynamic coefficient moment body y

_f

-axis ( · );

C

_n

= Aerodynamic coefficient moment body z

_f

-axis ( · );

C

_D

= Aerodynamic drag coefficient, force negative air-path x

_a

-axis ( · );

C

_L

= Aerodynamic lift coefficient, force negative air-path z

_a

-axis ( · );

C

_X

= Aerodynamic coefficient force body x

_f

-axis ( · );

C

_Y

= Aerodynamic coefficient force body y

_f

-axis (·);

C

_Z

= Aerodynamic coefficient force body z

_f

-axis (·);

D = Drag force, positive along negative air-path X

_a

-axis (L = −X

_a^A

, N);

F = Thrust force (N), positive along body x

_f

-axis;

L = Lift force, positive along negative air-path Z

_a

-axis ( L = −Z

_a^A

, N);

L

_f

= Roll moment (N m), mathematically positive around x

_f

-axis;

M

_f

= Pitch moment (N m), mathematically positive around y

_f

-axis;

N

_f

= Yaw moment (N m), mathematically positive around z

_f

-axis;

S = Wing area ( S = 0 . 550 m

²

);

V

_A

, V

_A

= Aircraft speed and velocity relative to air ( V

_A

= kV

_A

k

₂

, m s

⁻¹

);

X

^A

= Resulting force along air-path x

_a

-axis (N);

X

_f^A

= Aerodynamic force along body x

_f

-axis (N);

X

_f^F

= Thrust force along body x

_f

-axis (N);

Y

^A

= Resulting force along air-path y

_a

-axis (N);

Y

_f^A

= Aerodynamic force along body y

_f

-axis (N);

Z

^A

= Resulting force along air-path z

_a

-axis (N);

Z

_f^A

= Aerodynamic force along body z

_f

-axis (N);

(·)

^post

= Domain of high angle of attack;

(·)

^pre

= Domain of low angle of attack;

x

_a

, y

_a

, z

_a

= Air-path axis system;

x

_f

, y

_f

, z

_f

= Body axis system;

x

_g

, y

_g

, z

_g

= Normal earth-fixed axis system;