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Global chattering-free sliding mode trajectory tracking control of underactuated autonomous underwater vehicles
MA Limin
Navy Military Representative Office in Jinzhou, Jinzhou 121000, China
Abstract: To investigate the global trajectory tracking control problem of an underactuated autonomous underwater vehicle (AUV) with control input and velocity constraints, this study first linearized the kinematics to determine the commands of pseudo velocities and yaw angle. These commands solved the speed jump problem in the traditional backstepping method and ensured that the control input and velocity constraints were satisfied. In the second design of the dynamics, an adaptive chattering-free sliding mode technique was used to achieve the global trajectory tracking control of an underactuated AUV, which improved the essential flaws in the work by Yu that cannot guarantee yaw angle tracking. The robust adaptive sliding mode controller with bound estimation achieved enhanced performance for a general class of AUVs in the presence of possibly large parameter uncertainty and unknown environmental disturbances from a practical application viewpoint. Finally, complete stability analysis based on Lyapunov theorem and simulations demonstrated the robustness of the proposed controller to systematical uncertainties, as well as the global tracking ability of underactuated AUVs with control input and velocity constraints.
Key words: autonomous underwater vehicle     global control     sliding mode control     trajectory tracking     backstepping     adaptive     constraint     Lyapunov method

1 欠驱动AUV的运动建模

1)AUV运动学模型

 $$\left\{ \matrix{ \dot x = u\cos \psi - v\sin \psi \hfill \cr \dot y = u\sin \psi + v\cos \psi \hfill \cr \dot \psi = r \hfill \cr} \right.$$ (1)

2)AUV动力学模型

 $$\left\{ \matrix{ \dot u = {{{m_{22}}} \over {{m_{11}}}}vr - {{{d_{11}}} \over {{m_{11}}}}u + {1 \over {{m_{11}}}}({\tau _{d1}} + {\tau _u}) \hfill \cr \dot v = {{{m_{11}}} \over {{m_{22}}}}ur - {{{d_{22}}} \over {{m_{22}}}}v + {1 \over {{m_{22}}}}{\tau _{d2}} \hfill \cr \dot r = {{({m_{22}} - {m_{11}})} \over {{m_{33}}}}uv - {{{d_{33}}} \over {{m_{33}}}}r + {1 \over {{m_{33}}}}({\tau _{d3}} + {\tau _r}) \hfill \cr} \right.$$ (2)

2 轨迹跟踪控制器设计 2.1 虚拟参考信号的设计

 $$\left\{ \matrix{ {{\dot x}_d} = {u_d}\cos {\psi _d} - {v_d}\sin {\psi _d} \hfill \cr {{\dot y}_d} = {u_d}\sin {\psi _d} - {v_d}\cos {\psi _d} \hfill \cr {{\dot \psi }_d} = {r_d} \hfill \cr} \right.$$ (3)

 $${x_e} = x - {x_d},{y_e} = y - {y_d},{\psi _e} = \psi - {\psi _d}$$ (4)

 \eqalign{ & {{\dot x}_e} + {k_p}\tanh ({{\bar k}_p}{x_e}) = X \cr & {{\dot y}_e} + {k_p}\tanh ({{\bar k}_p}{y_e}) = Y \cr} (5)

 $$\left. {\left[ \matrix{ {e_x} \hfill \cr {e_y} \hfill \cr} \right.} \right]{\left[ {\matrix{ {\cos {\psi _d} - } & {\sin {\psi _d}} \cr {\sin {\psi _d}} & {\cos {\psi _d}} \cr } } \right]^{ - 1}}\left. {\left[ \matrix{ X \hfill \cr Y \hfill \cr} \right.} \right]$$ (6)
exey收敛到零，意味着xeye也收敛到零。根据式(6)，进一步整理得到
 $$\left. {\left[ \matrix{ {e_x} \hfill \cr {e_y} \hfill \cr} \right.} \right]\left[ {\matrix{ {\cos {\psi _e} - } & {\sin {\psi _e}} \cr {\sin {\psi _e}} & {\cos {\psi _e}} \cr } } \right]\left. {\left[ \matrix{ u \hfill \cr v \hfill \cr} \right.} \right] - \left. {\left[ \matrix{ {\bar X} \hfill \cr {\bar Y} \hfill \cr} \right.} \right]$$ (7)

\eqalign{ & \bar X = {u_d} - \cos {\psi _d}{k_p}\tanh ({{\bar k}_p}{x_e}) - \cr & sin{\psi _d}{k_p}\tanh ({{\bar k}_p}{y_e}) \cr & \bar Y = {u_d} + sin{\psi _d}{k_p}\tanh ({{\bar k}_p}{x_e}) - \cr & \cos {\psi _d}{k_p}\tanh ({{\bar k}_p}{y_e}) \cr}

 \eqalign{ & {\left[ {\matrix{ {\cos {\psi _e} - \sin {\psi _e}} \cr {\sin {\psi _e}\cos {\psi _e}} \cr } } \right]^{ - 1}}\left[ {\matrix{ {{e_x}} \cr {{e_y}} \cr } } \right] = \cr & \left[ {\matrix{ {u - \bar X\cos {\psi _e} - \bar Y\sin {\psi _e}} \cr {v + \bar X\sin {\psi _e} - \bar Y\cos {\psi _e}} \cr } } \right] \cr} (8)

 \eqalign{ & {u_{e = }}\cos {\psi _{ec}}\bar X + \sin {\psi _{ec}}\bar Y \cr & {v_e} = - \sin {\psi _{ec}}\bar X + \cos {\psi _{ec}}\bar Y \cr} (9)

 $${\psi _{ec}} = \theta - \varphi {\rm{ 或}}{\psi _{ec}} = \pi + \theta - \varphi {\rm{ }}$$ (10)

 \eqalign{ & {u_e} = cos{\psi _{ec}}\bar X + \sin {\psi _{ec}}\bar Y \cr & {\psi _c} = {\psi _{ec}} + {\psi _d} \cr & {\psi _{ec}} = \theta - \varphi 或{\psi _{ec}} = \pi + \theta - \varphi \cr} (11)

2.2 滑模控制器设计

 $${S_1} = {u_e} + {\lambda _1}{u_e} + {\lambda _2}\smallint {u_e}$$ (12)

$${\ddot u_e} + {{{\lambda _1}} \over {{m_{11}}}}({m_{22}}vr - {d_{11}}u + {\tau _{d1}} + {\tau _u} - {m_{11}}{u_e})$$

 \eqalign{ & {\tau _u} = {{\hat m}_{11}}({{\dot u}_c} - {{{{\ddot u}_e}} \over {{\lambda _1}}} - {{{\lambda _2}} \over {{\lambda _1}}}{u_e}) - {{\hat m}_{22}}ur + \cr & {{\hat d}_{11}}u + {{\hat \smallint }_1} - {k_1}{S_1} - {B_1} \cr} (14)

 $${B_i} = \left\{ \matrix{ {{\hat \delta }_i}{{{S_i}} \over {|{S_i}|}},{S_i} \ne 0 \hfill \cr 0,{S_i} = 0 \hfill \cr} \right.$$ (15)

 \begin{align} & {{{\dot{\hat{f}}}}_{1}}=-{{\Gamma }_{{{f}_{1}}}}{{S}_{1}}-{{\sigma }_{{{f}_{1}}}}{{\Gamma }_{{{f}_{1}}}}{{{\hat{f}}}_{1}} \\ & {{{\dot{\hat{\delta }}}}_{1}}={{\Gamma }_{{{\delta }_{1}}}}|{{S}_{1}}|-{{\sigma }_{{{\delta }_{1}}}}{{\Gamma }_{{{\delta }_{1}}}}|{{{\hat{\delta }}}_{1}}\text{ } \\ \end{align} (16)

 \eqalign{ & {\sigma _{fi}} = \left\{ \matrix{ 0,|{f_i}| \le {N_{fo}} \hfill \cr {\sigma _{f0}}({{|{f_i}|} \over {{N_{f0}}}} - 1),{N_{f0}} \le |{f_i}| \hfill \cr {\sigma _{f0}},|{f_i}| \ge 2{N_{f0}} \hfill \cr} \right. \le 2{N_{f0}} \cr & {\sigma _{\delta i}} = \left\{ \matrix{ 0,|{\delta _i}| \le {N_{\delta o}} \hfill \cr {\sigma _{\delta o}}({{|{\delta _i}|} \over {{N_{\delta o}}}} - 1),{N_{\delta 0}} \le |{\delta _i}| \hfill \cr {\sigma _{\delta o}},|{\sigma _i}| \ge 2{N_{\delta o}} \hfill \cr} \right. \le 2{N_{\delta 0}} \cr} (17)

 ${{V}_{1}}=\frac{1}{2{{\lambda }_{1}}}{{m}_{11}}S_{1}^{2}+\frac{1}{2}\Gamma _{{{f}_{1}}}^{-1}{{\tilde{f}}_{1}}^{2}+\frac{1}{2}\Gamma _{{{\delta }_{1}}}^{-1}{{\tilde{\delta }}_{1}}^{2}$ (18)

 $${\dot V_1} \le - {k_1}S_1^2 + {\sigma _{{f_1}}}{\tilde f_1}{\hat f_1} + {\sigma _{{\delta _1}}}{\tilde \delta _1}{\hat \delta _1}$$ (19)

 $${S_2} = {r_e} + {e_\psi } + {\lambda _3}\smallint ({r_e} + {e_\psi })$$ (20)

 \eqalign{ & {{\dot S}_2} = {1 \over {{m_{33}}}}[({m_{11}} - {m_{22}})uv - {d_{33}}r + {\tau _{d3}} + {\tau _r} - {m_{33}}{{\dot r}_d}] + \cr & {{\dot e}_\psi } + {\lambda _3}\smallint ({r_e} + {e_\psi }) \cr} (21)

 $${f_2} = ({m_{33}} - {\hat m_{33}})({\dot r_d} - {\dot e_\psi } - {\lambda _3}({r_e} + {e_\psi })$$ (22)

 \eqalign{ & {\tau _r} = {{\hat m}_{33}}[{{\dot r}_d} - {{\dot e}_\psi } - {\lambda _3}({r_e} + {e_\psi })] - ({{\hat m}_{11}} - {{\hat m}_{22}})uv + \cr & {{\hat d}_{33}}r + {{\hat f}_2} - {k_2}{S_2} - {B_2} \cr} (23)

 \begin{align} & {{{\dot{\hat{f}}}}_{2}}=-{{\Gamma }_{{{f}_{2}}}}{{S}_{2}}-{{\sigma }_{{{f}_{2}}}}{{\Gamma }_{{{f}_{2}}}}{{{\hat{f}}}_{2}} \\ & {{{\dot{\hat{\delta }}}}_{2}}={{\Gamma }_{{{\delta }_{2}}}}|{{S}_{2}}|-{{\sigma }_{{{\delta }_{2}}}}{{\Gamma }_{{{\delta }_{2}}}}{{{\hat{\delta }}}_{2}} \\ \end{align} (24)

 $${V_2} = {1 \over 2}{m_{33}}S_2^2 + {1 \over 2}\Gamma _{{f_2}}^{ - 1}{\tilde f_2}^2 + {1 \over 2}\Gamma _{{\delta _2}}^{ - 1}{\tilde \delta _2}^2$$ (25)

 $${\dot V_2} \le - {k_2}S_2^2 + {\sigma _{{f_2}}}{\tilde f_2}{\hat f_2} + {\sigma _{{\delta _2}}}{\tilde \delta _2}{\hat \delta _2}$$ (26)

3 稳定性分析

 \eqalign{ & {{\dot V}_3} = {{\dot V}_1} + {{\dot V}_2} \le \cr & - {k_1}S_2^2 + \sum\limits_{i = 1,2} {({\sigma _{{f_i}}}} {{\tilde f}_i}{{\hat f}_i} + {\sigma _{\delta i}}{{\tilde \delta }_i}{{\hat \delta }_i}) \cr} (27)

 \eqalign{ & {\sigma _{{f_i}}}{f_i}{{\hat f}_i} = {\sigma _{fi}}{{\tilde f}_i}({f_i} - {{\tilde f}_i}) \le \cr & - {\sigma _{fi}}{{\tilde f}^2}_i + {\sigma _{fi}}({{{{\tilde f}^2}_i + {f_i}^2} \over 2}) \le \cr & - {1 \over 2}{\sigma _{{f_i}}}{{\tilde f}^2}_i + {1 \over 2}{\sigma _{{f_i}}}{f_i}^2 \cr & {\sigma _{{\delta _i}}}{\delta _i}^T{{\hat \delta }^T}_i = ({\delta _i} - {{\hat \delta }_i}) \le \cr & - {\sigma _{{\delta _i}}}{{\tilde \delta }^2}_i + {\sigma _{{\delta _i}}}({{{{\tilde \delta }^2}_i + {\delta ^2}_i} \over 2}) \le \cr & - {1 \over 2}{\sigma _{\delta i}}{{\tilde \delta }^2}_i + {1 \over 2}{\sigma _{\delta i}}{\delta _i}^2 \cr} (28)

 \eqalign{ & {\sigma _{{f_i}}}{f_i}^2 \le 4{\sigma _{f0}}{N_{f0}} + {\sigma _{f0}}{f_i}^2 \cr & {\sigma _{{\delta _i}}}{\delta _i}^2 \le 4{\sigma _{{\delta _0}}}{N_{{\delta _0}}} + {\sigma _{{\delta _0}}}{\delta _i}^2 \cr} (29)

 \eqalign{ & {{\dot V}_3} \le - {k_1}S_1^2 - {k_2}S_2^2 - {1 \over 2}\sum\limits_{i = 1,2} {\left( {{\sigma _{{f_i}}}{{\tilde f}^2}_i + {\sigma _{{\delta _i}}}{{\tilde \delta }^2}_i} \right)} + \cr & \sum\limits_{i = 1,2} {\left( {{\sigma _{{f_0}}}{f^2}_i + {\sigma _{{\delta _0}}}{\delta ^2}_i} \right)} + 4{\sigma _{{\beta _0}}}{N_{{f_0}}} + 4{\sigma _{{\delta _0}}}{N_{{\delta _0}}} \le \cr & - \mu {V_3} + C \cr & \mu = \min \{ 2{k_1}{\lambda _1},2{k_2},{{{\sigma _{{f_i}}}} \over {{\lambda _{\max }}({\Gamma _{fi}}^{ - 1})}},{{{\sigma _{{\delta _i}}}} \over {{\lambda _{\max }}({\Gamma _{\delta i}}^{ - 1})}}\} \cr & C = 4{\sigma _{fo}}{N_{{f_0}}} + 4{\sigma _{\delta o}}{N_{\delta 0}} + \sum\limits_{i = 1,2} {\left( {{\sigma _{{f_0}}}{f^2}_i + {\sigma _{{\delta _0}}}{\delta ^2}_i} \right)} \cr} (30)

 $$0 \le {V_3}\left( t \right) \le {V_3}\left( 0 \right){e^{ - w}} + C/\mu$$ (31)

1)速度u的有界性:根据上述分析，速度u在控制器τu下可实现速度跟踪，即u=uc，所以虚拟速度控制量uc有界，即可保证速度u的有界性。根据式(11)得到，

 \eqalign{ & |{u_c}| = |cos{\psi _{ec}}\bar X + sin{\psi _{ec}}\bar Y| \le \cr & |{u_d}| + |{v_d}| + \sqrt {2{k_p}} \cr} (32)

2)速度r的有界性:根据控制器设计，角速度r在控制器τr下可实现速度跟踪，即r=rd，而参考轨迹是有界的，所以角速度r有界。

3)速度v的有界性:根据AUV动力学模型(2)，速度ur和扰动项τd2均有界，所以速度v有界。

4)横向速度跟踪误差ve的收敛性:定义横向速度误差ve=v-vc。根据虚拟控制量航向角ψec=θ-φ，或ψec=π+θ-φ可知：

 $$\tan {\psi _{ec}} = {{\bar Yu - \bar Xv} \over {\bar Xu + \bar Yv}}$$ (33)

 $$v = {v_e} + {T_u}{u_e}$$ (34)

4 仿真实验结果与分析

 图 1 圆形轨迹跟踪Fig. 1 Circular trajectory-tracking

 图 2 位置和航向跟踪误差Fig. 2 Tracking errors of position and orientation

 图 3 速度跟踪响应曲线Fig. 3 Response curves of velocity-tracking

 图 4 控制输入响应曲线Fig. 4 Response curves of control inputs

 图 5 正弦轨迹跟踪Fig. 5 Sinusoidal trajectory-tracking

 图 6 位置和航向跟踪误差Fig. 6 Tracking errors of position and orientation

 图 7 速度跟踪响应曲线Fig. 7 Response curves of velocity-tracking

 图 8 控制输入响应曲线Fig. 8 Response curves of control inputs
5 结束语

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DOI: 10.11992/tis.201512015

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#### 文章信息

MA Limin

Global chattering-free sliding mode trajectory tracking control of underactuated autonomous underwater vehicles

CAAI Transactions on Intelligent Systems, 2016, 11(02): 200-207.
DOI: 10.11992/tis.201512015