﻿ 悬挂负载空中机器人的抗摆控制
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 应用科技  2020, Vol. 47 Issue (2): 17-22  DOI: 10.11991/yykj.201906003 0

### 引用本文

CHANG Le, YANG Zhong, ZHANG Qiuyan, et al. Anti-swing control research of aerial robot with suspended load[J]. Applied Science and Technology, 2020, 47(2): 17-22. DOI: 10.11991/yykj.201906003.

### 文章历史

1. 南京航空航天大学 自动化学院，江苏 南京 211106;
2. 贵州电网有限责任公司电力科学研究院，贵州 贵阳 550002

Anti-swing control research of aerial robot with suspended load
CHANG Le1, YANG Zhong1, ZHANG Qiuyan2, WANG Shaohui1, LI Jiewen1
1. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;
2. Electric Power Research Institute of Guizhou Power Grid Co., Ltd, Guiyang 550002, China
Abstract: In this paper, in order to solve the problem of load swing and residual oscillation during the flight of an aerial robot with suspended load, the sliding mode variable structure method is used to design the height controller and anti-swing controller. Firstly, the dynamic model of an aerial robot with suspended load is established. On this basis, the height controller of the first-order sliding mode and the anti-swing controller of the second-order sliding mode are designed. An improved boundary layer method with variable thickness of boundary layer is proposed to reduce the chattering phenomenon. Finally, the validity of the controller is verified by the joint simulation with MATLAB and Adams. The results show that the designed controller has excellent anti-swing effect and meets the system requirements.
Keywords: aerial robot    unmanned air vehicle    suspended load    anti-swing control    sliding mode control    hierarchical sliding-mode    integral back-stepping    variable boundary layer

1 数学模型

 $\left\{ {\begin{array}{*{20}{c}} {{F_X} = - F\sin \theta } \\ {{F_Z} = F\cos \theta } \end{array}} \right.$

 $\left\{ {\begin{array}{*{20}{c}} {{x_L} = {x_Q} + L\sin \alpha } \\ {{z_L} = {z_Q} - L\cos \alpha } \end{array}} \right.$ (1)

 $\left\{ {\begin{array}{*{20}{c}} {{{\ddot x}_L} = {{\ddot x}_Q} - L\dot \alpha \sin \alpha + L\ddot \alpha \cos \alpha } \\ {{{\ddot z}_L} = {{\ddot z}_Q} + L\dot \alpha \cos \alpha + L\ddot \alpha \sin \alpha } \end{array}} \right.$ (2)

 $\left\{ \begin{array}{l} {m_Q}{{\ddot x}_Q} = {F_X} + T\sin \alpha \\ {m_Q}{{\ddot z}_Q} = {F_Z} - T\cos \alpha - {m_Q}g \\ {m_L}{{\ddot x}_L} = - T\sin \alpha \\ {m_L}{{\ddot z}_L} = T\cos \alpha - {m_L}g \\ \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} T = \dfrac{{{m_L}( - {F_X}\sin \alpha + {F_Z}\cos \alpha + L{M_Q}{{\dot \alpha }^2})}}{{{m_Q} + {m_L}}} \\ \ddot \alpha = - \dfrac{1}{{{m_Q}L}}({F_X}\cos \alpha + {F_Z}\sin \alpha ) \\ \end{array} \right.$ (4)

 $\left\{ {\begin{array}{*{20}{l}} {\begin{array}{*{20}{c}} {{{\ddot x}_Q} = \dfrac{{{m_Q} + {m_L}{{\cos }^2}\alpha }}{{({m_Q} + m){m_Q}}}{F_X} + \dfrac{{{m_L}\sin \alpha \cos \alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z} + } \\ {{\rm{ }}\dfrac{{{m_L}L}}{{{m_Q} + {m_L}}}{{\dot \alpha }^2}\sin \alpha {\rm{ }}} \end{array}} \\ {{{\ddot z}_Q} = \dfrac{{{m_L}\sin \alpha \cos \alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_X} + \dfrac{{{m_Q} + {m_L}{{\sin }^2}\alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z} - } \\ \quad \quad \quad {\dfrac{{{m_L}L}}{{{m_Q} + {m_L}}}{{\dot \alpha }^2}\cos \alpha - g{\rm{ }}} \\ {\ddot \alpha = - \dfrac{{\cos \alpha }}{{{m_L}L}}{F_X} - \dfrac{{\sin \alpha }}{{{m_L}L}}{F_Z}{\rm{ }}} \end{array}} \right.$ (5)
2 控制器设计

 $\left\{ {\begin{array}{*{20}{l}} {\dfrac{{{m_L}\sin \alpha \cos \alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z} \ll \dfrac{{{m_Q} + {m_L}{{\cos }^2}\alpha }}{{({m_Q} + m){m_Q}}}{F_X}} \\ {\dfrac{{{m_L}\sin \alpha \cos \alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_X} \ll \dfrac{{{m_Q} + {m_L}{{\sin }^2}\alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z}} \\ {\dfrac{{\sin \alpha }}{{{m_L}L}}{F_Z} \ll \dfrac{{\cos \alpha }}{{{m_L}L}}{F_X}} \end{array}} \right.$

 $\left\{ \begin{array}{l} {{\ddot x}_Q} = \dfrac{{{m_Q} + {m_L}{{\cos }^2}\alpha }}{{({m_Q} + m){m_Q}}}{F_X} + \dfrac{{{m_L}L}}{{{m_Q} + {m_L}}}{{\dot \alpha }^2}\sin \alpha \\ {{\ddot z}_Q} = \dfrac{{{m_Q} + {m_L}{{\sin }^2}\alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z} - \dfrac{{{m_L}L}}{{{m_Q} + {m_L}}}{{\dot \alpha }^2}\cos \alpha - g \\ \ddot \alpha = - \dfrac{{\cos \alpha }}{{{m_L}L}}{F_X} \end{array} \right.$

2.1 高度控制器设计

 ${\ddot z_Q} = \frac{{{m_Q} + {m_L}{{\sin }^2}\alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_Z} - \frac{{{m_L}L}}{{{m_Q} + {m_L}}}{\dot \alpha ^2}\cos \alpha - g$

 $\left\{ \begin{array}{l} {x_2} = {{\dot x}_1} \\ {{\dot x}_2} = {f_1}({{x}}) + {b_1}({{x}}){F_Z} \\ \end{array} \right.$

 ${s_1} = {c_1}{e_z} + {\dot e_z}$

${s_1}$ 求导可得：

 ${\dot s_1} = {c_1}{\dot x_1} + {\dot x_2} = {c_1}{x_2} + {f_1}({{x}}) + {b_1}({{x}}){F_Z}$

 ${V_1} = \frac{1}{2}{s_1}^2$ (6)

 ${\dot V_1} = {s_1}({c_1}{x_2} + {f_1}({{x}}) + {b_1}({{x}}){F_Z})$

 $- {\varepsilon _1}{\rm{sgn}} ({s_1}) - {K_1}{s_1} = {c_1}{x_2} + {f_1}({{x}}) + {b_1}({{x}}){F_Z}$

 ${F_Z} = \frac{{ - {c_1}{x_2} - {f_1}({{x}}) - {\varepsilon _1}{\rm{sgn}} ({s_1}) - {K_1}{s_1}}}{{{b_1}({{x}})}}$ (7)

 ${\dot V_1} = - {\varepsilon _1}\left| {{s_1}} \right| - {K_1}{s_1}^2 \leqslant 0$

2.2 抗摆控制器设计

 $\left\{ {\begin{array}{*{20}{l}} {{{\ddot x}_Q} = \dfrac{{{m_Q} + {m_L}{{\cos }^2}\alpha }}{{({m_Q} + {m_L}){m_Q}}}{F_X} + \dfrac{{{m_L}L}}{{{m_Q} + {m_L}}}{{\dot \alpha }^2}\sin \alpha } \\ {\ddot \alpha = - \dfrac{{\cos \alpha }}{{{m_L}L}}{F_X}} \end{array}} \right.$ (8)

 $\left\{ \begin{array}{l} {x_4} = {{\dot x}_3} \\ {{\dot x}_4} = {f_2}({{x}}) + {b_2}({{x}}){F_X} \\ {x_6} = {{\dot x}_5} \\ {{\dot x}_6} = {f_3}({{x}}) + {b_3}({{x}}){F_X} \\ \end{array} \right.$

 $A:\left\{ {\begin{array}{*{20}{l}} {{x_2} = {{\dot x}_1}} \\ {{{\dot x}_2} = {f_1}({{x}}) + {b_1}({{x}}){F_X}} \end{array}} \right.$
 $B:\left\{ {\begin{array}{*{20}{l}} {{x_4} = {{\dot x}_3}} \\ {{{\dot x}_4} = {f_2}({{x}}) + {b_2}({{x}}){F_X}} \end{array}} \right.$

 ${s_2} = {c_2}{e_x} + {\dot e_x}$
 ${s_3} = {c_3}{e_\alpha } + {\dot e_\alpha }$

 ${u_{{\rm{eq}}_{\rm{1}}}} = - \frac{{{c_1}{x_2} + {f_1}({{x}})}}{{{b_1}({{x}})}}$
 ${u_{{\rm{eq}}_2}} = - \frac{{{c_2}{x_4} + {f_2}({{x}})}}{{{b_2}({{x}})}}$

 ${F_X} = {u_{{\rm{e}}{{\rm{q}}_{\rm{1}}}}} + {u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{{\rm{SW}}}}$

 $S = {a_1}{s_2} + {a_2}{s_3}$

 $V = \frac{1}{2}{S^2}$

$V$ 求导可得：

 $\begin{split} \dot V = & S\dot S = S[{a_1}({c_1}{{\dot x}_1} + {{\dot x}_2}) + {a_2}({c_2}{{\dot x}_3} + {{\dot x}_4})] = \\ & {\rm{ }}S[{a_1}({c_1}{x_2} + {f_1} + {b_1}({u_{{\rm{e}}{{\rm{q}}_{\rm{1}}}}} + {u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{{\rm{SW}}}})) + \\ & {\rm{ }}{a_2}({c_2}{x_4} + {f_2} + {b_2}({u_{{\rm{e}}{{\rm{q}}_{\rm{1}}}}} + {u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{{\rm{SW}}}}))] = \\ & {\rm{ }}S[{a_1}{b_1}({u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{{\rm{SW}}}}) + {a_2}{b_2}({u_{{\rm{e}}{{\rm{q}}_{\rm{1}}}}} + {u_{{\rm{SW}}}})] \end{split}$

 $- {\varepsilon _2}{\rm{sgn}} (S) - {K_2}S = {a_1}{b_1}({u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{SW}}) + {a_2}{b_2}({u_{{\rm{e}}{{\rm{q}}_1}}} + {u_{{\rm{SW}}}})$ (9)

 $\begin{array}{l} {u_{{\rm{SW}}}} = {({a_1}{b_1} + {a_2}{b_2})^{ - 1}}[ - {a_2}{b_2}{u_{{\rm{e}}{{\rm{q}}_1}}} - {a_1}{b_1}{u_{{\rm{e}}{{\rm{q}}_2}}} - \\ \quad {\rm{ }}{\varepsilon _2}{\rm{sgn}} (S) - {K_2}S] \end{array}$

 $\begin{split} & \quad \quad \quad \quad \quad {F_X} = {u_{{\rm{e}}{{\rm{q}}_{\rm{1}}}}} + {u_{{\rm{e}}{{\rm{q}}_2}}} + {u_{SW}} = \\ & {u_{{\rm{e}}{{\rm{q}}_1}}} + {u_{{\rm{e}}{{\rm{q}}_2}}} + \dfrac{{ - {a_2}{b_2}{u_{{\rm{e}}{{\rm{q}}_1}}} - {a_1}{b_1}{u_{{\rm{e}}{{\rm{q}}_2}}} - {\varepsilon _2}{\rm{sgn}} (S) - {K_2}S}}{{{a_1}{b_1} + {a_2}{b_2}}} = \\ & \dfrac{{{a_1}{b_1}}}{{{a_1}{b_1} + {a_2}{b_2}}}{u_{{\rm{e}}{{\rm{q}}_1}}} + \dfrac{{{a_2}{b_2}}}{{{a_1}{b_1} + {a_2}{b_2}}}{u_{{\rm{e}}{{\rm{q}}_2}}} + \dfrac{{ - {\varepsilon _2}{\rm{sgn}} (S) - {K_2}S}}{{{a_1}{b_1} + {a_2}{b_2}}} \end{split}$ (10)

 $\dot V = S( - {\varepsilon _2}{\rm{sgn}} (S) - {K_2}S) = - {\varepsilon _2}\left| S \right| - {K_2}{S^2} \leqslant 0$ (11)

2.3 边界层设计

 ${\rm{sat}}(S) = \left\{ {\begin{array}{*{20}{c}} {1{\rm{ , }}S > \mathit{\Delta} } \\ {kS{\rm{ , }}\left| S \right| \leqslant \mathit{\Delta} } \\ { - 1{\rm{ , }}S < \mathit{\Delta} } \end{array}} \right.,k = 1/\mathit{\Delta}$

 $\mathit{\Delta} {\rm{ = }}\left\{ {\begin{array}{*{20}{l}} {{\mathit{\Delta} _1},{\rm{ }}\left| e \right| < {e_{\min }}} \\ {{\mathit{\Delta} _2},{\rm{ }}\left| e \right| > {e_{\max }}} \\ {\dfrac{{({e_{\max }} - \left| e \right|){\mathit{\Delta} _1} + (\left| e \right| - {e_{\min }}){\mathit{\Delta} _2}}}{{{e_{\max }} - {e_{\min }}}},\;\;\;{\text{其他}}} \end{array}} \right.$

3 仿真结果与分析

4 结论

1）采用分层滑模控制方法能够解决驱动量少于自由度的控制问题，值得其他欠驱动系统推广使用；

2）本文针对悬挂负载空中机器人的二维模型进行了控制器设计，仿真结果说明了控制算法的有效性，可以考虑将本方法进一步运用到三维模型中；

3）本文设计的边界层厚度可变的边界层法能够有效降低滑模控制存在的抖震现象，值得深入研究；

4）仿真结果表明，本文所设计的控制器能够将负载摆角始终控制在±5°以内，控制效果较好，具有实际应用价值，可应用于货物运输等领域。