The translation dynamics are derived in the kinematic coordinate system (
K) for which the
\(x_K\)-axis points into the direction of the kinematic velocity of the aircraft,
, and the position propagation is given in the local coordinate system (
N):
$$\begin{aligned} m\dot{V}_K&= P_V - D - mg\sin \gamma _K, \end{aligned}$$
(1)
$$\begin{aligned} m V_K\dot{\gamma }_K&= P_{\gamma } + L - mg\cos \gamma _K, \end{aligned}$$
(2)
$$\begin{aligned} \dot{h}_N&= V_K\sin \gamma _K. \end{aligned}$$
(3)
In the above equations,
\(P_V\) and
\(P_{\gamma }\) denote the thrust forces,
m is the aircraft mass,
g the gravitational constant, and
D and
L represent the drag and lift forces, respectively. The last two variables are defined in the kinematic coordinate system (
K) as follows:
$$\begin{aligned} \begin{pmatrix} D \\ L \\ \end{pmatrix} = \begin{bmatrix} \cos (\alpha _A-\alpha _K)&\quad -\sin (\alpha _A-\alpha _K) \\ \sin (\alpha _A-\alpha _K)&\quad \cos (\alpha _A-\alpha _K) \\ \end{bmatrix} \cdot \begin{pmatrix} \hat{D} \\ \hat{L} \\ \end{pmatrix}, \end{aligned}$$
(4)
where
\(\hat{D}\) and
\(\hat{L}\) are, respectively, the drag and lift forces in the aerodynamic reference system (
A). They are given by the relations
$$\begin{aligned} \hat{D}&= \frac{1}{2}\,f_D(\alpha _A, M) \rho (h) \textit{SV}_A^2, \end{aligned}$$
(5)
$$\begin{aligned} \hat{L}&= \frac{1}{2}f_L(\alpha _A, M) \rho (h) \textit{SV}_A^2, \end{aligned}$$
(6)
where
\(\alpha _{A}\) is the aerodynamic angle of attack,
M the Mach number,
\(\rho (h)\) the air density at altitude
h,
S the wing area, and
\(V_{A}\) is the absolute value of the aerodynamic velocity. The lift and drag coefficients
\(f_D(\alpha _A, M)\) and
\(f_L(\alpha _A, M)\) are taken in the form:
$$\begin{aligned} f_D(\alpha _A, M)&= c_1^D+c_2^D \alpha _A+ c_3^D M+c_4^D \alpha _A^2+c_5^D\alpha _A M\nonumber \\&\quad +c_6^D M^2+c_7^D\alpha _A^3 +c_8^D\alpha _A^2 M+c_9^D\alpha _A M^2, \end{aligned}$$
(7)
$$\begin{aligned} f_L(\alpha _A, M)&= c_1^L+c_2^L \alpha _A+c_3^L M+c_4^L \alpha _A^2+c_5^L\alpha _A M\nonumber \\&\quad +c_6^L M^2+c_7^L\alpha _A^3+c_8^L\alpha _A^2 M+c_9^L\alpha _A M^2, \end{aligned}$$
(8)
where the constants
\(c_i^D\) and
\(c_i^L\),
\(i=1,\ldots ,9\) are found from least square fitting to experimental data. The absolute value of the aerodynamic velocity
\(V_{A}\) can be derived using its relation to the kinematic velocity
considered in the local frame (
N) and the wind velocities
\(W_{x}\) and
\(W_{h}\) in the
\(x_N\)- and
\(h_N\)-direction, respectively:
Therefore,
$$\begin{aligned} V_A=\left[ (V_K \cos \gamma _K -W_x)^2 + (V_K \sin \gamma _K -W_h)^2\right] ^{1/2}. \end{aligned}$$
The Mach number
M is defined as the ratio between the absolute value of the aerodynamic velocity
\(V_\mathrm{A}\) and the speed of sound
c:
$$\begin{aligned} M=\frac{V_A}{c}, \quad c = \sqrt{\kappa \textit{RT}(h)}, \end{aligned}$$
(10)
with
c depending on the adiabatic index for air,
\(\kappa \), the gas constant for ideal gases,
R, and the temperature of air,
T(
h), at the altitude
h.
From the formulas of the International Standard Atmosphere ISA (DIN ISO 2533) for the troposphere layer (
\(h = -2 \ldots 11\,\text {km}\), relative to sea level), the temperature of air,
T(
h), and air density,
\(\rho (h)\), are approximated as follows:
$$\begin{aligned} T(h)&= T_\mathrm{s} \cdot \Bigl [1-\frac{n-1}{n}\frac{g}{R \cdot T_\mathrm{s}}\cdot H_\mathrm{G} \Bigr ], \end{aligned}$$
(11)
$$\begin{aligned} \rho (h)&= \rho _\mathrm{s} \cdot \Bigl [1-\frac{n-1}{n}\frac{g}{R \cdot T_\mathrm{s}}\cdot H_\mathrm{G} \Bigr ]^{\frac{1}{n-1}}, \end{aligned}$$
(12)
$$\begin{aligned} H_\mathrm{G}&= \frac{r_\mathrm{E}\cdot h}{r_\mathrm{E} + h}. \end{aligned}$$
(13)
In the formulas above,
n is the polytropic exponent,
g the gravitational constant,
\(r_\mathrm{E}\) the earth radius, and
\(T_\mathrm{s}\) and
\(\rho _\mathrm{s}\) are the reference temperature and density of air, respectively.
For modeling thrust forces, a two engine setup with thrust inclination angle
\(\sigma \) is considered. The following three main components influencing the thrust can be referred: the gross thrust (GT), the ram drag (RD), and the cowl drag (CD). The net thrust components
\(P_V\) and
\(P_{\gamma }\), appearing in Eqs. (
1) and (
2), are defined as follows in the kinematic frame (
K):
$$\begin{aligned} \begin{pmatrix} P_V \\ P_{\gamma } \\ \end{pmatrix} = \begin{bmatrix} \cos \alpha _K&\quad \sin \alpha _K \\ -\sin \alpha _K&\quad \cos \alpha _K \\ \end{bmatrix} \cdot \begin{pmatrix} \hat{P}_V \\ \hat{P}_{\gamma } \\ \end{pmatrix}, \end{aligned}$$
(14)
where the components
\(\hat{P}_V\) and
\(\hat{P}_{\gamma }\) are computed by subtracting the drag components RD and CD from the gross thrust GT in the body-fixed frame (
B):
$$\begin{aligned} \hat{P}_V&= 2 \cdot [\hbox {GT} \cdot \cos \sigma - (\hbox {RD} + \hbox {CD}) \cdot \cos \alpha _A], \end{aligned}$$
(15)
$$\begin{aligned} \hat{P}_{\gamma }&= 2 \cdot [ \hbox {GT} \cdot \sin \sigma - (\hbox {RD} + \hbox {CD}) \cdot \sin \alpha _A]. \end{aligned}$$
(16)
Here, the thrust components GT, RD, and CD are approximated by a second-order polynomial least squares fit depending on the thrust command
\(\delta _T\in [0,1]\) and the Mach number
M using the constants
\(c_i^{\mathrm{GT}}\),
\(c_i^{\mathrm{RD}}\), and
\(c_i^{\mathrm{CD}}\)
\(i=1,\ldots ,5\):
$$\begin{aligned} \hbox {GT}(\delta _T, M)&= c_1^{\mathrm{GT}}+c_2^{\mathrm{GT}} M + c_2^{\mathrm{GT}} \delta _T+c_3^{\mathrm{GT}} M^2 + c_4^{\mathrm{GT}} \delta _T M + c_5^{\mathrm{GT}} \delta _T^2, \end{aligned}$$
(17)
$$\begin{aligned} \hbox {RD}(\delta _T, M)&= c_1^{\mathrm{RD}}+c_2^{\mathrm{RD}} M + c_2^{\mathrm{RD}} \delta _T+c_3^{\mathrm{RD}} M^2 + c_4^{\mathrm{RD}} \delta _T M + c_5^{\mathrm{RD}} \delta _T^2, \end{aligned}$$
(18)
$$\begin{aligned} \hbox {CD}(\delta _T, M)&= c_1^{\mathrm{CD}}+c_2^{\mathrm{CD}} M + c_2^{\mathrm{CD}} \delta _T+c_3^{\mathrm{CD}} M^2 + c_4^{\mathrm{CD}} \delta _T M + c_5^{\mathrm{CD}} \delta _T^2. \end{aligned}$$
(19)
Introduce the following additional equations that smooth the controls:
$$\begin{aligned} \dot{\alpha }_K = \widetilde{\alpha }_K,\quad \dot{\delta }_T = \widetilde{\delta }_T. \end{aligned}$$
(20)
The following equations smooth the wind disturbances:
$$\begin{aligned} \dot{W_x} = -k_\mathsf{w} \left( W_x-\widetilde{W}_x\right) ,\quad \dot{W_h} = -k_\mathsf{w} \left( W_h-\widetilde{W}_h\right) ,\quad \text {with}\quad k_\mathsf{w}=1\,1/\mathrm{s}. \end{aligned}$$
(21)