![](https://csdnimg.cn/release/download_crawler_static/87486109/bg7.jpg)
机翼产生的升力$F_L^i({\theta _f},{v_x},{v_z})$和阻力$F_D^i({\theta _f},
{v_x},{v_z})$为:
$$\left[ {\begin{array}{*{20}{c}} {F_D^i} \\ 0 \\ {F_L^i} \end{array}} \right] = {\boldsymbol{R}}({\theta _f})\left[ {\begin{array}{*{20}{c}} { -
\tfrac{1}{2}{C_D}{\theta _f}\rho Av_f^2} \\ 0 \\ { - \tfrac{1}{2}{C_L}{\theta _f}\rho Av_f^2} \end{array}} \right]$$
其中, $i = 1,2$分别代表左右两个机翼; $\rho $是空气密度, $A$是机翼面积, ${v_f}$是
气流速度, ${\boldsymbol{R}}({\theta _f})$是绕$y$轴把机翼上的力分解到机体轴上的旋转矩
阵; ${C_D}$, ${C_L}$分别表示机翼的阻力和升力系数; ${v_f} = (v_x^2 \; + v_z^2)^{{1 /
2}}$, ${v_x}$和${v_z}$分别为机体在${x_b}$, ${z_b}$轴方向的速度; 设左右两机翼的攻角
相同, 产生的升阻力相同.
变体无人机的动态模型可转化为更加紧凑的非线性系统形式:
$$\left\{ \begin{aligned} &{{{\dot{\boldsymbol X}}}_1} = {{\boldsymbol{X}}_2} \\ &{{{\dot{\boldsymbol X}}}_2} = {\boldsymbol{f}} +
{\boldsymbol{B\varsigma }} + {\boldsymbol{d}} \end{aligned} \right.$$
其中, ${{\boldsymbol{X}}_1} ={[{{\boldsymbol{\xi }}_{3 \times
1}^{\rm{T}}},{{\boldsymbol{\eta }}_{3 \times
1}^{\rm{T}}}]^{\rm{T}}}$, ${{\boldsymbol{X}}_2}= {[{{\dot{\boldsymbol \xi }}_{3 \times
1}^{\rm{T}}},{{\dot{\boldsymbol \eta }}_{3 \times
1}^{\rm{T}}}]^{\rm{T}}}$, ${\boldsymbol{\varsigma }}= [{{\boldsymbol{\tau }}_{3 \times
1}^{\rm{T}}}, {{\boldsymbol{u}}_{3 \times 1}^{\rm{T}}}]^{\rm{T}} = {[{\tau _1}, {\tau _2},
{\tau _3}, {u_2},{u_3}, {u_4}]^{\rm{T}}}$, ${\boldsymbol{B}} =
{\rm{diag}}\{ {\boldsymbol{B}}_{\xi 3 \times 3}^{\rm{T}}, {{\boldsymbol{B}}_{\eta 3 \times
3}^{\rm{T}}}\} = {\rm{diag\;\{ }}{1 / m},{1 / m},{1 / m},$${1 / {{I_{xx}},}}{1 /
{{I_{yy}}}},{1 / {{I_{zz}}}}\} $, ${\boldsymbol{d}} = {[{\boldsymbol{d}}_{\xi 3 \times
1}^{\rm{T}},{\boldsymbol{d}}_{\eta 3 \times 1}^{\rm{T}}]^{\rm{T}}} =
{[{d_1},\,{d_2},\,{d_3},\,{d_4},\,{d_5},\,{d_6}]^{\rm{T}}}$为外部扰动,
$$\begin{split} {\boldsymbol{f}} =& \left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{f}}_{\xi 3 \times 1}}} \\ {{{\boldsymbol{f}}_{\eta 3 \times 1}}}
\end{array}} \right]= \\ &\left[ {\begin{array}{*{20}{l}} {{\dfrac{{{W_x}}} m}} \\ {{\dfrac{{{W_y}}} m}} \\ {g + {\dfrac{{{W_z}}} {m}}} \\
\dfrac{\left( {{I_{yy}} - {I_{zz}}} \right)\dot \theta \dot \psi } {{{I_{xx}}}} - \dfrac{{{I_r}\dot \theta {\omega _q}}}{ {{I_{xx}}}} \\
\dfrac{ {\left( {{I_{zz}} - {I_{xx}}} \right)\dot \phi \dot \psi + {I_r}(\dot \phi {\omega _q} + \dot \psi {\omega _r}) + {W_t}} } {{I_{yy}}} \\
\dfrac{\left( {{I_{xx}} - {I_{yy}}} \right)\dot \phi \dot \theta } {{{I_{zz}}}} - \dfrac{{I_r}\dot \theta {\omega _r}} {{I_{zz}}} \end{array}}
\right]\end{split}$$
飞行器需要跟踪所需的理想姿态角, 并使总推力达到所需的加速度. 为了计算所需的
理想姿态角和总推力, 本文采用动态反演的方法, 通过将虚拟控制输入${\tau _1}$, ${\tau
_2}$, ${\tau _3}$ 等价于位置动力学, 得到以下方程: