Angles

atack angle
$$\alpha$$

trajectory angle
$$\gamma$$

roll angle
$$\phi$$

pitch angle
$$\theta$$

yaw angle
$$\psi$$

slid angle
$$\beta$$

Rates

roll rate
$$p$$

pitch rate
$$q$$

yaw rate
$$r$$

Speed

Speed vector
$$V = \sqrt{u^2 + v^2 + w^2}$$

longitudinal speed vector
$$u = V \cos{\beta} \cos{\alpha}$$

lateral speed vector
$$v = V \sin(\beta)$$

vertical speed vector
$$w = V \cos{\beta} \sin{\alpha}$$

Quaternions

$$\eta_0 = \cos(\frac{\phi}{2}) \cos(\frac{\theta}{2}) \cos(\frac{\psi}{2}) + \sin(\frac{\phi}{2}) \sin(\frac{\theta}{2}) \sin(\frac{\psi}{2})$$

$$\eta_1 = \sin(\frac{\phi}{2}) \cos(\frac{\theta}{2}) \cos(\frac{\psi}{2}) + \cos(\frac{\phi}{2}) \sin(\frac{\theta}{2}) \sin(\frac{\psi}{2})$$

$$\eta_2 = \cos(\frac{\phi}{2}) \sin(\frac{\theta}{2}) \cos(\frac{\psi}{2}) + \sin(\frac{\phi}{2}) \cos(\frac{\theta}{2}) \sin(\frac{\psi}{2})$$

$$\eta_3 = \cos(\frac{\phi}{2}) \cos(\frac{\theta}{2}) \sin(\frac{\psi}{2}) + \sin(\frac{\phi}{2}) \sin(\frac{\theta}{2}) \cos(\frac{\psi}{2})$$

$$\begin{bmatrix} \dot{\eta_0} \\ \dot{\eta_1} \\ \dot{\eta_2} \\ \dot{\eta_3} \\ \end{bmatrix} = \frac{1}{2} \begin{bmatrix} -\eta_1 & -\eta_2 & -\eta_3 \\ \eta_0 & -\eta_3 & \eta_2 \\ \eta_3 & \eta_0 & -\eta_1 \\ -\eta_2 & \eta_1 & \eta_0 \\ \end{bmatrix} \begin{bmatrix} p \\ q \\ r \\ \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 0 & -p & -q & -r \\ p & 0 & r & -q \\ q & -r & 0 & p \\ r & q & -p & 0 \\ \end{bmatrix} \begin{bmatrix} \eta_0 \\ \eta_1 \\ \eta_2 \\ \eta_3 \\ \end{bmatrix}$$

Stabiliy Derivatives
$$C_{L_0}$$

$$C_{L_\alpha}$$

$$C_{L_{\dot{\alpha}}}$$

$$C_{L_q}$$

$$C_{D_0}$$

$$C_{D_\alpha}$$

$$C_{D_{\dot{\alpha}}}$$

$$C_{D_q}$$

$$C_{m_0}$$

$$C_{m_\alpha}$$

$$C_{m_{\dot{\alpha}}}$$

$$C_{m_q}$$

$$C_{y_\beta}$$

$$C_{l_\beta}$$

$$C_{l_p}$$

$$C_{l_r}$$

$$C_{n_\beta}$$

$$C_{n_p}$$

$$C_{n_r}$$

Control Derivatives
$$C_{L_{\delta_{e}}}$$

$$C_{D_{\delta_{e}}}$$

$$C_{y_{\delta_{\alpha}}}$$

$$C_{y_{\delta_{r}}}$$

$$C_{l_{\delta_{a}}}$$

$$C_{l_{\delta_{r}}}$$

$$C_{y_{\delta_{a}}}$$

$$C_{m_{\delta_{e}}}$$

$$C_{n_{\delta_{a}}}$$

$$C_{n_{\delta_{r}}}$$

General equations

$$C_L = \frac{L}{\rho V S}$$

$$C_D = \frac{D}{\rho V S}$$

$$E = \frac{L}{D} = \frac{C_L}{C_D}$$

$$F = m a$$

$$P = F V$$

$$P = J t$$

Forces Coeficients

Lift

$$C_L = C_{L_0} + C_{L_\alpha} \alpha + \frac{\bar{c}}{2V} (C_{L_{\dot{\alpha}}} \dot{\alpha} + C_{L_q} q) + C_{L_{\delta_{e}}} \delta_{e}$$

Drag

$$C_D = C_{D_0} + C_{D_\alpha} \alpha + \frac{\bar{c}}{2V} (C_{D_{\dot{\alpha}}} \dot{\alpha} + C_{D_q} q) + C_{D_{\delta_{e}}} \delta_{e}$$

Lateral Force

$$C_y = C_{y_\beta} \beta + C_{y_{\delta_{\alpha}}} \delta_{\alpha} + C_{y_{\delta_{r}}} \delta{r}$$

Moments Coeficients

Roll

$$C_l = C_{l_\beta} \beta + \frac{b}{2V} (C_{l_p} p + C_{l_r} r) + C_{l_{\delta_{a}}} \delta_{a} + C_{l_{\delta_{r}}} \delta_{r}$$

Pitch
$$C_m = C_{m_0} + C_{m_\alpha} \alpha + \alpha + \frac{\bar{c}}{2V} (C_{m_{\dot{\alpha}}} \dot{\alpha} + C_{m_q} q) + C_{m_{\delta_{e}}} \delta_{e}$$

Yaw
$$C_n = C_{n_\beta} \beta + \frac{b}{2V} (C_{n_p} p + C_{n_r} r) + C_{n_{\delta_{a}}} \delta_{a} + C_{n_{\delta_{r}}} \delta_{r}$$