六自由度运动方程（Six-DOF Motion Equations）

$$\begin{bmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{bmatrix} = \mathbf{R}_{EB} \begin{bmatrix} u \\ v \\ w \end{bmatrix}$$

$$\begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix} = \begin{bmatrix} 1 & \sin\phi\tan\theta & \cos\phi\tan\theta \\ 0 & \cos\phi & -\sin\phi \\ 0 & \sin\phi\sec\theta & \cos\phi\sec\theta \end{bmatrix} \begin{bmatrix} p \\ q \\ r \end{bmatrix}$$

$$m(\dot{u} + qw - rv - g\sin\theta) = X$$

$$m(\dot{v} + ru - pw + g\cos\theta\sin\phi) = Y$$

$$m(\dot{w} + pv - qu + g\cos\theta\cos\phi) = Z$$

$$I_x \dot{p} - (I_y - I_z)qr = L$$

$$I_y \dot{q} - (I_z - I_x)rp = M$$

$$I_z \dot{r} - (I_x - I_y)pq = N$$

$$L_{static} = (\rho_{air} - \rho_{He}) V_{He} g$$

$$B = L_{static} - mg$$

$$M_a = \begin{bmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{bmatrix}$$

$$m_{11} = \frac{1}{3}\rho V \frac{b^2}{a^2 + b^2}$$

$$x_{n+1} = x_n + \frac{\Delta t}{6}(k_1 + 2k_2 + 2k_3 + k_4)$$

$$\sum F = 0, \quad \sum M = 0$$