建立无人机系统如上图所示。无人机的姿态角与四个电机有如下关系：
ϕ = + M 1 + M 2 − M 3 − M 4 (roll) \phi = + M_1 + M_2 - M_3 - M_4 \tag{roll} ϕ=+M1+M2−M3−M4(roll) θ = − M 1 + M 2 + M 3 − M 4 (pitch) \theta = - M_1 + M_2 + M_3 - M_4 \tag{pitch} θ=−M1+M2+M3−M4(pitch) ψ = − M 1 + M 2 − M 3 + M 4 (yaw) \psi = - M_1 + M_2 - M_3 + M_4 \tag{yaw} ψ=−M1+M2−M3+M4(yaw)
写成矩阵的形式为：
[ ϕ θ ψ ] = [ + 1 + 1 − 1 − 1 − 1 + 1 + 1 − 1 − 1 + 1 − 1 + 1 ] [ M 1 M 2 M 3 M 4 ] \left[\begin{matrix} \phi \\ \theta \\ \psi \\ \end{matrix}\right]= \left[\begin{matrix} +1 & +1 & -1 & -1 \\ -1 & +1 & +1 & -1 \\ -1 & +1 & -1 & +1 \\ \end{matrix}\right] \left[\begin{matrix} M_1 \\ M_2 \\ M_3 \\ M_4 \\ \end{matrix}\right] ⎣
⎡ϕθψ⎦
⎤=⎣
⎡+1−1−1+1+1+1−1+1−1−1−1+1⎦
⎤⎣
⎡M1M2M3M4⎦
⎤
利用矩阵求逆运算，转换一下，有如下关系：
[ M 1 M 2 M 3 M 4 ] = [ + 1 − 1 − 1 + 1 + 1 + 1 − 1 + 1 − 1 − 1 − 1 + 1 ] [ ϕ θ ψ ] \left[\begin{matrix} M_1 \\ M_2 \\ M_3 \\ M_4 \\ \end{matrix}\right]= \left[\begin{matrix} +1 & -1 & -1 \\ +1 & +1 & +1 \\ -1 & +1 & -1 \\ -1 & -1 & +1 \\ \end{matrix}\right] \left[\begin{matrix} \phi \\ \theta \\ \psi \\ \end{matrix}\right] ⎣
⎡M1M2M3M4⎦
⎤=⎣
⎡+1+1−1−1−1+1+1−1−1+1−1+1⎦
⎤⎣
⎡ϕθψ⎦
⎤
展开后可以得到：
M 1 = + ϕ − θ − ψ (M1) M_1 = +\phi - \theta - \psi \tag{M1} M1=+ϕ−θ−ψ(M1) M 2 = + ϕ + θ + ψ (M2) M_2 = +\phi + \theta + \psi \tag{M2} M2=+ϕ+θ+ψ(M2) M 3 = − ϕ + θ − ψ (M3) M_3 = -\phi + \theta - \psi \tag{M3} M3=−ϕ+θ−ψ(M3) M 4 = − ϕ − θ + ψ (M4) M_4 = -\phi - \theta + \psi \tag{M4} M4=−ϕ−θ+ψ(M4)