Cheat Sheet for dynamic equation of aircraft

기호의 정의는 다음과 같다.

v : 선형 속도 벡터

$\omega$ : 회전 속도 벡터

F : 힘 벡터

M : 모멘트 벡터

g : 중력 가속도 벡터

$C_N^B$ : 항법 좌표계에서 동체축 좌표계로 좌표계를 변환하는 방향 코사인 행렬 (Direction Cosine Matrix)

 

위첨자 B, N, T는 순서대로 다음과 같다.

B : 동체축 좌표계에서 정의된 벡터

N : 항법 좌표계에서 정의된 벡터

T : 행렬의 전치

 

 

Force and Moment equation on Flat earth model

$$F^B = m\left( \omega^B \times v^B \right) - m C_N^B g^N $$

$$M^B = J \left( \dot{\omega}^B + \omega \times I \omega \right)$$

 

True airspeed 진 대기속도 $V_T$, Angle of Attack(AOA) 받음각 $\alpha$, Angle of Sideslip 옆미끄럼각(AOS) $\beta$

$$v^B = \left[ \matrix{U & V & W }\right]^T$$

$$V_T = \sqrt{U^2 + V^2 + W^2}$$

$$\alpha = \text{atan}(W/U) = \text{atan2}(W, U)$$

$$\beta = \text{asin}(V/V_T)$$

 

Derivatives of states

Force and Moment equations

$$\dot{v}^B = - \left( \omega^B \times v^B \right) + C_N^B g^N + F^B /m$$

$$\dot{\omega}^B = J^{-1}\left(-\omega^B \times I \omega + M^B \right)$$

 

For attitude

Euler angle

$$\left[\matrix{ \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} }\right] = \left[ \matrix{ 1 & \text{sin}\phi \text{tan}\theta & \text{cos}\phi \text{tan}\theta \\ 0 & \text{cos}\phi & - \text{sin}\phi \\ 0 & \text{sin}\phi /\text{cos}\theta & \text{cos}\phi / \text{cos}\theta } \right] \left[ \matrix{P \\ Q \\R} \right]$$

Quaternion

$$\dot{q} = \frac{1}{2}\begin{bmatrix}0 & -\left(\omega^B \right)^T \\ \omega^B & -\Omega^B \end{bmatrix} \begin{bmatrix} q_0 \\ \textbf{q} \end{bmatrix}$$

$$\begin{bmatrix}\dot{q_0} \\ \dot{q_1} \\ \dot{q_2} \\ \dot{q_3} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 0 & -p & -q & -r \\ p & 0 & r & -q \\ q & -r & 0 & p \\ r & q & -p & 0\end{bmatrix} \begin{bmatrix} q_0 \\ q_1 \\ q_2 \\ q_3 \end{bmatrix}$$

Direction Cosine Matrix(DCM)

$$\frac{d C_b^a}{dt} =- \Omega_{ab}^b C_b^a =\begin{bmatrix}0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{bmatrix} C_b^a $$

Aerodynamic States

$$\dot{V_T} = \frac{U\dot{U} + V\dot{V} + W\dot{W}} {V_T}$$

$$\dot{\alpha} = \frac{u\dot{w} - \dot{u}w}{U^2 + W^2 }$$

$$\dot{\beta} = \frac{\dot{V}V_T - V \dot{V_T} } {V_T^2\text{cos}\beta} = \frac{\dot{V}V_T - V \dot{V_T} } {V_T \sqrt{U^2 + W^2 }}$$

 

 

Transformation between frames

$$C_n^b = C_\phi C_\theta C_\psi$$

$$r^{b} = \begin{bmatrix}1 & 0 & 0 \\ 0 & \text{cos}\phi & \text{sin}\phi \\0 & -\text{sin}\phi & \text{cos}\phi \end{bmatrix} \begin{bmatrix} \text{cos}\theta & 0 & -\text{sin}\theta \\ 0 & 1 & 0 \\ \text{sin}\theta & 0 & \text{cos}\theta \end{bmatrix} \begin{bmatrix} \text{cos}\psi & \text{sin}\psi & 0 \\ -\text{sin}\psi & \text{cos}\psi & 0 \\ 0 & 0 & 1\end{bmatrix} r^n $$

$$C_n^b = \begin{bmatrix} c\theta c\psi & c\theta s\psi & -s\theta \\ \left(-c\phi s\psi + s\phi s \theta c\psi \right) & \left(c\phi c \psi +s\phi s\theta s\psi \right) & s \phi c\theta \\ \left(s\phi s\psi +c\phi s \theta c\psi \right) & \left(-s\phi c\psi + c\phi s\theta s\psi \right) & c\phi c\theta\end{bmatrix} = \left(C_b^n \right)^T$$

On WGS84 Earth Model..

Eccentriccity $e=0.8181919$

Flattening $f=0.003352810664 = 1/298.257223563$

Meridian Radius $R_M = \frac{R(1-e^2)}{\left(1-e^2 sin^2 L \right)^{3/2}}$

Equatorial Radius $R_N = \frac{R}{\left(1-e^2 sin ^2 L \right)^{1/2}}$

$$p^e = \begin{bmatrix} X^e \\ Y^e \\ Z^e \end{bmatrix} =\begin{bmatrix} \left(R_N +h \right) cos(L) cos(l) \\ \left(R_N + h \right) cos(L) sin(l) \\ \left(R_N (1-e^2) + h \right) sin (L)\end{bmatrix}$$

Between geodetic and navigation frame(Flat earth)

$$\begin{bmatrix} X^n \\ Y^n \\ Z^n \end{bmatrix} = \begin{bmatrix} \frac{L - L_0}{atan(1/R_M)} \\ \frac{l - l_0 } {atan(1/(R_N cos(L))} \\ -(h-h_0) \end{bmatrix} $$

$$\begin{bmatrix}L\\l\\h \end{bmatrix} = \begin{bmatrix} L_0 + X^n atan(1/R_N)  \\ l_0 + Y^n atan(1/R_M cos(L))\\ h_0 -Z^n  \end{bmatrix}$$

$$\begin{bmatrix} \dot{L} \\ \dot{l} \\ \dot{h} \end{bmatrix} = \begin{bmatrix} \frac{V^N}{R_N + h} \\ \frac{V^E} {(R_M +h ) cos(L)} \\ - V_D \end{bmatrix}$$

Kinematics equations

$$\left[ \matrix{P \\ Q \\R} \right] = \left[ \matrix{1 & 0 & -\text{sin}\theta \\ 0 & \text{cos}\phi & \text{sin}\phi \text{cos}\theta \\ 0 & -\text{sin} \phi & \text{cos}\phi \text{cos}\theta} \right] \left[\matrix{ \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} }\right]$$

$$\left[\matrix{ \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} }\right] = \left[ \matrix{ 1 & \text{sin}\phi \text{tan}\theta & \text{cos}\phi \text{tan}\theta \\ 0 & \text{cos}\phi & - \text{sin}\phi \\ 0 & \text{sin}\phi /\text{cos}\theta & \text{cos}\phi / \text{cos}\theta } \right] \left[ \matrix{P \\ Q \\R} \right]$$

미분계수의 유도

$$\text{tan}\alpha = W/U$$

$$\frac{1}{\text{cos}^2 \alpha } \dot{\alpha}= \frac{u\dot{w} - \dot{u}w}{u^2}$$

$\text{cos}^2 \alpha = \frac{U^2 + W^2 }{U^2}$ 이므로

$$\therefore) \hspace{3mm}\dot{\alpha} = \frac{u\dot{w} - \dot{u}w}{U^2 + W^2 }$$

 

$$\text{sin} \beta = V/V_T$$

$$\dot{\beta}\text{cos} \beta = \frac{\dot{V}V_T - V \dot{V_T} } { V_T^2}$$

$\text{cos}^2 \beta = \left(U^2 + W^2 \right) /V_T^2$ 이므로

$$\therefore) \hspace{3mm} \dot{\beta} = \frac{\dot{V}V_T - V \dot{V_T} } {V_T^2\text{cos}\beta} = \frac{\dot{V}V_T - V \dot{V_T} } {V_T \sqrt{U^2 + W^2 }}$$