﻿ 一种小型倾转四旋翼飞行器的轨迹控制
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 应用科技  2018, Vol. 45 Issue (3): 71-75  DOI: 10.11991/yykj.201706014 0

### 引用本文

SHEN Yangyang, YANG Zhong, XU Hao, et al. Trajectory control for a small quad tilt-rotor aircraft[J]. Applied Science and Technology, 2018, 45(3), 71-75. DOI: 10.11991/yykj.201706014.

### 文章历史

Trajectory control for a small quad tilt-rotor aircraft
SHEN Yangyang, YANG Zhong, XU Hao, LI Jinsong, YANG Qing
School of Automation, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Abstract: Considering the influence of external disturbances, modeling mismatches and input delays in the position control of small quad tilt-rotor, the dynamic model of quad tilt-rotor(QTR) was given firstly. In addition, a control method based on disturbance observer was applied to design the position controller of QTR. The robust servomechanism linear quadratic regulator (RSLQR) was applied as the attitude controller of inner loop. A flight scenario was designed in the numerical simulation to verify the trajectory tracking performance of the QTR, the result shows that the proposed position controller based on disturbance observer can realize the tracking of the desired trajectory, QTR can not only take into account the functions of the traditional rotor UAV, but also can fly for a long distance like the fixed-wing aircraft.
Key words: tilt-rotor aircraft    disturbance observer    rslqr    riccati equation    disturbance    modeling error    time delay    integral filter

1 QTR动力学模型

 $\left[ {\begin{array}{*{20}{c}}{m{{{I}}_{3 \times 3}}} & {{{{O}}_{3 \times 3}}}\\{{{{O}}_{3 \times 3}}} & {{{{I}}_b}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{{\dot{ V}}}_w}}\\{{{{\dot{ \Omega }}}_b}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0\\{{{{\Omega }}_b} \times \left( {{{{I}}_b}{{{\Omega }}_b}} \right)}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{{{F}}_t}}\\{{{{M}}_t}}\end{array}} \right]$

 \left\{ \begin{aligned}& \ddot X = \frac{1}{m}\left[ {\left( {{c_\psi }{c_\theta }{c_{{\theta _w}}} - \left( {{c_\phi }{s_\theta }{c_\psi } + {s_\phi }{s_\psi }} \right){s_{{\theta _w}}}} \right){u_1} + {W_x} - {k_x}\dot X\left| {\dot X} \right|} \right]\\& \ddot Y = \frac{1}{m}\left[ {\left( {{s_\psi }{c_\theta }{c_{{\theta _w}}} - \left( {{c_\phi }{s_\theta }{s_\psi } - {s_\phi }{c_\psi }} \right){s_{{\theta _w}}}} \right){u_1} + {W_y} - {k_y}\dot Y\left| {\dot Y} \right|} \right]\\& \ddot Z = \frac{1}{m}\left[ {\left( { - {s_\theta }{c_{{\theta _w}}} - {c_\phi }{c_\theta }{s_{{\theta _w}}}} \right){u_1} + mg + {W_z} - {k_z}\dot Z\left| {\dot Z} \right|} \right]\\& \dot p = \frac{{{s_{{\theta _w}}}{u_2} - {c_{{\theta _w}}}{u_4}}}{{{I_x}}} + \frac{{{I_y} - {I_z}}}{{{I_x}}}qr - \frac{{{J_{{\rm{prop}}}}}}{{{I_x}}}q{\omega _p}{s_{{\theta _w}}}\\& \dot q = \frac{{{s_{{\theta _w}}}{u_3}}}{{{I_y}}} + \frac{{{I_z} - {I_x}}}{{{I_y}}}pr + \frac{{{J_{{\rm{prop}}}}}}{{{I_y}}}\left( {p{s_{{\theta _w}}} + r{c_{{\theta _w}}}} \right){\omega _p}\\& \dot r = \frac{{{c_{{\theta _w}}}{u_2} + {s_{{\theta _w}}}{u_4}}}{{{I_z}}} + \frac{{{I_x} - {I_y}}}{{{I_z}}}pq - \frac{{{J_{{\rm{prop}}}}}}{{{I_z}}}q{\omega _p}{c_{{\theta _w}}}\end{aligned} \right. (1)

2 基于扰动观测器的位置控制 2.1 系统不确定项问题描述

 $\left\{ \begin{array}{l}{\dot {{P}}} = {{{V}}_w}\\{{{\dot{ V}}}_w} = F({{x}}) + G({{x}})u(t - \tau ) + d({{x}},t)\end{array} \right.$ (2)

 $F({{x}}){\rm{ = }}{F_n}({{x}}) + {F_\Delta }({{x}}),G({{x}}){\rm{ = }}{G_n}({{x}}) + {G_\Delta }({{x}})$ (3)
2.2 位置控制

 $\left\{ \begin{array}{l}\ddot X = \displaystyle\frac{1}{m}\left[ {\left( {{c_\psi }{c_{{\theta _w}}} - \left( {{c_\phi }{c_\psi }\theta + {s_\psi }\phi } \right){s_{{\theta _w}}}} \right){u_1} + {W_x} - {k_x}\dot X\left| {\dot X} \right|} \right]\\[5pt]\ddot Y = \displaystyle\frac{1}{m}\left[ {\left( {{s_\psi }{c_{{\theta _w}}} - \left( {{c_\phi }{s_\psi }\theta - {c_\psi }\phi } \right){s_{{\theta _w}}}} \right){u_1} + {W_y} - {k_y}\dot Y\left| {\dot Y} \right|} \right]\\[5pt]\ddot Z = \displaystyle\frac{1}{m}\left[ {\left( { - {c_{{\theta _w}}}\theta - {c_\theta }{s_{{\theta _w}}}} \right){u_1} + mg + {W_z} - {k_z}\dot Z\left| {\dot Z} \right|} \right]\end{array} \right.$

 ${{{F}}_n}({{x}}) = \frac{1}{m}\left[ {\begin{array}{*{20}{c}} {{W_x} - {k_x}\dot X\left| {\dot X} \right|} \\[7pt] {{W_y} - {k_y}\dot Y\left| {\dot Y} \right|} \\[7pt] {{W_z} + mg - {k_z}\dot Z\left| {\dot Z} \right|} \end{array}} \right]$ (4)
 ${{{G}}_n}({{x}}) = \frac{1}{m}\left[ {\begin{array}{*{20}{c}} {{c_\psi }{s_{{\theta _w}}}{u_1}}&{{s_\psi }{s_{{\theta _w}}}{u_1}}&{{c_\psi }{c_{{\theta _w}}}} \\[9pt] {{s_\psi }{s_{{\theta _w}}}{u_1}}&{{c_\psi }{s_{{\theta _w}}}{u_1}}&{{s_\psi }{c_{{\theta _w}}}} \\[9pt] { - {c_{{\theta _w}}}}&0&{{c_\phi }{c_\theta }{s_{{\theta _w}}}} \end{array}} \right]$ (5)

 ${{u}} = {\left[ {\theta ,\phi ,{u_1}} \right]^{\rm{T}}}$ (6)

 ${e_1} = {{P}} - {{{P}}_r},\;\;{e_2} = {{{V}}_w} - {\alpha _1}$ (7)

 ${\dot e_1} = {e_2} + {\alpha _1} - {{\dot{ P}}_r}$ (8)

 ${\alpha _1}{\rm{ = }} - {k_{1p}}{e_1} - {k_{1i}}\int\limits_0^t {{e_1}{\rm{d}}t} + {\dot {{P}}_r}$ (9)

 ${\dot \alpha _1}{\rm{ = }} - {k_{1p}}{\dot e_1} - {k_{1i}}{e_1} + {{\ddot{ P}}_r} = - {k_{1p}}\left( {{V_w} - {{{\dot{ P}}}_r}} \right) - {k_{1i}}{e_1} + {{\ddot{ P}}_r}$

 $\begin{split}{{\dot e}_2} = & {{{\dot{ V}}}_w} - {{\dot \alpha }_1} = {{F}}({{x}}) + {{G}}({{x}})u\left( {t - \tau } \right) + d({{x}},t) - {{\dot \alpha }_1} = \\& {{{F}}_n}({{x}}) + {{{G}}_n}({{x}})u - {{\dot \alpha }_1} + d({{x}},t) + {{{F}}_\Delta }({{x}}) + {{{G}}_\Delta }({{x}})u\end{split}$ (10)

 $\begin{split}& w = d({{x}},t) + {{{F}}_\Delta }({{x}}) + {{{G}}_\Delta }({{x}})u(t - \tau ) = \\& \quad \quad {{\dot e}_2} - ({{{F}}_n}({{x}}) + {{{G}}_n}({{x}})u - {{\dot \alpha }_1})\end{split}$ (11)

 $\tilde w = \frac{{{e_2} + {\alpha _1} - \displaystyle\int\limits_0^t {\left( {{{{F}}_n}({{x}}) + {{{G}}_n}({{x}})u(t)} \right){\rm{d}}t} }}{t}$ (12)

 $u = \frac{{{{\dot e}_2} + {{\dot \alpha }_1} - {{{F}}_n}({{x}}) - \tilde w}}{{{{{G}}_n}({{x}})}}$

 $u = \frac{{ - {k_2}{e_2} + {{\dot \alpha }_1} - {{{F}}_n}({{x}}) - \tilde w}}{{{{{G}}_n}({{x}})}}$ (13)

2.3 位置控制稳定性证明

 $\left\{ \begin{array}{l}{{\dot e}_0} = {e_1}\\[8pt]{{\dot e}_1} = {e_2} - {k_{1p}}{e_1} - {k_{1i}}{e_0}\\[8pt]{{\dot e}_2} = - {k_2}{e_2}\end{array} \right.$

 \begin{aligned}\dot V = & 2a{k_{1i}}{e_0}{{\dot e}_0} + 2a{e_1}{{\dot e}_1} + 2b{e_2}{{\dot e}_2}{\rm{ = }}\\& 2a{k_{1i}}{e_0}{e_1} + 2a{e_1}{e_2} - 2{k_{1p}}{e_1}^2 - 2a{k_{1i}}{e_0}{e_1} - \\& 2b{k_2}{e_2}^2 = - 2a{k_{1p}}{e_1}^2 + 2a{e_1}{e_2} - 2b{k_2}{e_2}^2{\rm{ = }}\\& - a{({e_1} - {e_2})^2} - a(2{k_{1p}} - 1){e_1}^2 - (2b{k_2} - a){e_2}^2\end{aligned}

 $2{k_{1p}} - 1 \geqslant 0, \;\; 2b{k_2} - a \geqslant 0$

2.4 姿态控制

 $\dot { {z}} = {\tilde{ A}}{ z} + {\tilde{ B}}\mu ,{ z} = \left[ {\begin{array}{*{20}{c}} e \\ {\dot x} \end{array}} \right],\mu = {\dot u_2}$

3 仿真实验

4 结论

1) 设计了一种倾转旋翼无人机QTR；

2) 考虑了控制模型中的不确定项，基于扰动观测器设计了位置控制器，并证明了其稳定性；

3) 采用了鲁棒伺服线性二次型控制理论设计了姿态控制器。