四轴模型在空中飞行，需要满足牛顿第二定律和欧拉运动定律：
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\begin{cases} m\overrightarrow{a}= m\cfrac{d\overrightarrow{V}}{dt} = m \overrightarrow{g} + M \overrightarrow{L} \\ \overrightarrow{G} + \overrightarrow{\tau} = I\overrightarrow{\dot{\omega}} + \overrightarrow{\omega} \times I \overrightarrow{\omega} \end{cases}
⎩⎨⎧ma=mdtdV=mg+MLG+τ=Iω˙+ω×Iω
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ω：飞控模型的角速度，
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\overrightarrow{\omega^b}=\overrightarrow{\omega^b_x}+\overrightarrow{\omega^b_y}+\overrightarrow{\omega^b_z}
ωb=ωxb+ωyb+ωzb
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\overrightarrow{V}
V：飞机模型的速度
G
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\overrightarrow{G}
G：陀螺力矩
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\tau
τ：螺旋桨在机体轴上产生的力矩
I：3×3的惯性矩阵
T：每个电机和桨叶提供的升力
L：模型自身动力（升力），
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\overrightarrow{L}=\overrightarrow{T_1}+\overrightarrow{T_2}+\overrightarrow{T_3}+\overrightarrow{T_4}
L=T1+T2+T3+T4
注：四旋翼飞行器动力学及运动学方程参考，这个也重新从牛顿第二定律温习推倒了一遍。
位置动力学模型
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\begin{cases} \dot{V^e_{x} }= -\cfrac{L}{m}\cdot(\cos\phi\cdot\sin\theta\cdot\cos\psi + \sin\phi\cdot\sin\psi) \\ \dot{V^e_{y}} = -\cfrac{L}{m}\cdot(\cos\phi\cdot\sin\theta\cdot\sin\psi - \sin\phi\cdot\cos\psi) \\ \dot{V^e_{z}} = g-\cfrac{L}{m}\cdot(\cos\phi\cdot\cos\theta) \end{cases}
⎩⎨⎧Vxe˙=−mL⋅(cosϕ⋅sinθ⋅cosψ+sinϕ⋅sinψ)Vye˙=−mL⋅(cosϕ⋅sinθ⋅sinψ−sinϕ⋅cosψ)Vze˙=g−mL⋅(cosϕ⋅cosθ)
姿态动力学模型
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\begin{cases} \dot{\omega^b_x}(\phi)= \cfrac{1}{I_{xx}}[\tau_x + \omega^b_y\omega^b_z(I_{yy} - I_{zz}) - I_{RP\omega^b_y}\varOmega] \\ \dot{\omega^b_y}(\theta)= \cfrac{1}{I_{yy}}[\tau_y + \omega^b_x\omega^b_z(I_{zz} - I_{xx}) + I_{RP\omega^b_x)}\varOmega] \\ \dot{\omega^b_z}(\psi)= \cfrac{1}{I_{zz}}[\tau_z + \omega^b_x\omega^b_y(I_{xx} - I_{yy})] \end{cases}
⎩⎨⎧ωxb˙(ϕ)=Ixx1[τx+ωybωzb(Iyy−Izz)−IRPωybΩ]ωyb˙(θ)=Iyy1[τy+ωxbωzb(Izz−Ixx)+IRPωxb)Ω]ωzb˙(ψ)=Izz1[τz+ωxbωyb(Ixx−Iyy)]
姿态角的变化率：
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W=\begin{bmatrix} 1 & \tan(\theta)\sin(\phi) & \tan(\theta)\cos(\phi) \\ 0 & \cos(\phi) & -\sin(\phi) \\ 0 & \sin(\phi)/\cos(\theta) & \cos(\phi)/\cos(\theta) \end{bmatrix}
W=⎣⎡100tan(θ)sin(ϕ)cos(ϕ)sin(ϕ)/cos(θ)tan(θ)cos(ϕ)−sin(ϕ)cos(ϕ)/cos(θ)⎦⎤
在小扰动的情况下，即各个角度的变化较小的前提下，姿态角的变化率与机体的旋转角速度近似相等，则有：
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\begin{bmatrix} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{bmatrix}=\begin{bmatrix} \omega^b_x \\ \omega^b_y \\ \omega^b_z \end{bmatrix}
⎣⎡ϕ˙θ˙ψ˙⎦⎤=⎣⎡ωxbωybωzb⎦⎤
飞控刚体模型
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\begin{cases} \ddot{x^e }= -\cfrac{L}{m}\cdot(\cos\phi\cdot\sin\theta\cdot\cos\psi + \sin\phi\cdot\sin\psi) \\ \ddot{y^e} = -\cfrac{L}{m}\cdot(\cos\phi\cdot\sin\theta\cdot\sin\psi - \sin\phi\cdot\cos\psi) \\ \ddot{z^e} = g-\cfrac{L}{m}\cdot(\cos\phi\cdot\cos\theta) \\ \ddot{\phi}= \cfrac{1}{I_{xx}}[\tau_x + \omega^b_y\omega^b_z(I_{yy} - I_{zz}) - I_{RP\omega^b_y}\varOmega] \\ \ddot{\theta}= \cfrac{1}{I_{yy}}[\tau_y + \omega^b_x\omega^b_z(I_{zz} - I_{xx}) + I_{RP\omega^b_x)}\varOmega] \\ \ddot{\psi}= \cfrac{1}{I_{zz}}[\tau_z + \omega^b_x\omega^b_y(I_{xx} - I_{yy})] \end{cases}
⎩⎨⎧xe¨=−mL⋅(cosϕ⋅sinθ⋅cosψ+sinϕ⋅sinψ)ye¨=−mL⋅(cosϕ⋅sinθ⋅sinψ−sinϕ⋅cosψ)ze¨=g−mL⋅(cosϕ⋅cosθ)ϕ¨=Ixx1[τx+ωybωzb(Iyy−Izz)−IRPωybΩ]θ¨=Iyy1[τy+ωxbωzb(Izz−Ixx)+IRPωxb)Ω]ψ¨=Izz1[τz+ωxbωyb(Ixx−Iyy)]
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