海洋科学  2020, Vol. 44 Issue (9): 130-138 PDF
http://dx.doi.org/10.11759/hykx20190821002

#### 文章信息

LI Xiao-gang, WANG Hong-du, LI Ming, LIU Xin. 2020.

Sliding mode active disturbance rejection control of underwater vehicle-manipulator system

Marina Sciences, 44(9): 130-138.
http://dx.doi.org/10.11759/hykx20190821002

### 文章历史

1. 中国海洋大学 工程学院, 山东 青岛 266100;
2. 四川警学院 侦查系, 四川 泸州 646000

Sliding mode active disturbance rejection control of underwater vehicle-manipulator system
LI Xiao-gang1, WANG Hong-du1, LI Ming1, LIU Xin2
1. Ocean University of China, College of Engineering, Qingdao 266100, China;
2. Sichuan Police College, Department of Investigation, Luzhou, Sichuan 646000, China
Abstract: To improve the control performance of underwater vehicle-manipulator systems (UVMSs), a sliding mode active disturbance rejection controller (SM-ADRC) is proposed wfor dealing with issues of the strong coupling, nonlinearity, and complex multi-source marine environment interference of UVMSs. The internal parameter uncertainties, measurement error, modeling error, and ocean current disturbance are regarded as the total disturbances, and the linear extended state observer is designed to estimate and attenuate this disturbance. Moreover, in this case, the complex system model is transformed into a simple integrated series system. Considering that the sliding mode controller (SMC) is insensitive to parameter perturbation, the SMC is incorporated into the control system to enhance the anti-disturbance performance of the whole system, and the stability of the control system is analyzed using the Lyapunov theory. The simulation results show that compared with the conventional SMC and linear active disturbance rejection controller, the UVMS with SMADRC schemes can achieve better trajectory tracking and anti-disturbance ability.
Key words: underwater vehicle manipulator system    active disturbance rejection control    linear extended state observer    sliding mode control

1 UVMS动力学分析 1.1 UVMS的动力学模型

 $\mathit{\boldsymbol{M}}(\mathit{\boldsymbol{q}})\mathit{\boldsymbol{\ddot q}} + \mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{D}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}})\mathit{\boldsymbol{\dot q}} + \mathit{\boldsymbol{G}}(\mathit{\boldsymbol{q}}) + \mathit{\boldsymbol{F}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}) = {\mathit{\boldsymbol{\tau }}_{\rm{c}}} + {\mathit{\boldsymbol{\tau }}_{\rm{d}}}$ (1)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{M}}(\mathit{\boldsymbol{q}}) = {\mathit{\boldsymbol{M}}^{\rm{T}}}(\mathit{\boldsymbol{q}}), \mathit{\boldsymbol{M}}(\mathit{\boldsymbol{q}}) > 0, \forall \mathit{\boldsymbol{q}} \in {\mathit{\boldsymbol{R}}^{6 + n}}\\ {\mathit{\boldsymbol{s}}^{\rm{T}}}{\rm{[}}\mathit{\boldsymbol{\dot M}}(\mathit{\boldsymbol{q}}) - 2\mathit{\boldsymbol{C}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}})]\mathit{\boldsymbol{s}} = 0, \forall \mathit{\boldsymbol{s}} \in {\mathit{\boldsymbol{R}}^{6 + n}}, \forall \mathit{\boldsymbol{q}} \in {\mathit{\boldsymbol{R}}^{6 + n}}\\ \mathit{\boldsymbol{D}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}) > 0, \forall \mathit{\boldsymbol{q}} \in {\mathit{\boldsymbol{R}}^{6 + n}} \end{array} \right.$ (2)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\hat M}}(\mathit{\boldsymbol{q}}) = {{\mathit{\boldsymbol{\hat M}}}^{\rm{T}}}(\mathit{\boldsymbol{q}}), \mathit{\boldsymbol{\hat M}}(\mathit{\boldsymbol{q}}) > 0, \forall \mathit{\boldsymbol{q}} \in {\mathit{\boldsymbol{R}}^{6 + n}}\\ {\mathit{\boldsymbol{s}}^{\rm{T}}}[\mathit{\boldsymbol{\dot {\hat M}}}(\mathit{\boldsymbol{q}}) - 2\mathit{\boldsymbol{\hat C}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}})]\mathit{\boldsymbol{s}} = 0, \forall \mathit{\boldsymbol{s}} \in {\mathit{\boldsymbol{R}}^{6 + n}}, \forall \mathit{\boldsymbol{q}} \in {\mathit{\boldsymbol{R}}^{6 + n}} \end{array} \right.$ (6)

 $\begin{array}{l} \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1m}}}\\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2m}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{a_{m1}}}&{{a_{m2}}}& \cdots &{{a_{mm}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\ddot y}_{_1}}}\\ {{{\ddot y}_{_2}}}\\ \vdots \\ {{{\ddot y}_m}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{c_{11}}}&{{c_{12}}}& \cdots &{{c_{1m}}}\\ {{c_{21}}}&{{c_{22}}}& \cdots &{{c_{2m}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{c_{m1}}}&{{c_{m2}}}& \cdots &{{c_{mm}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\dot y}_1}}\\ {{{\dot y}_2}}\\ \vdots \\ {{{\dot y}_m}} \end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}} {{u_1}}\\ {{u_2}}\\ \vdots \\ {{u_m}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {{f_1}}\\ {{f_2}}\\ \vdots \\ {{f_m}} \end{array}} \right] \end{array}$ (7)

 图 1 UVMS的动态模型示意图 Fig. 1 Schematic of the dynamic model of UVMS
1.2 水动力参数分析

 ${F_{{\rm{drag}}}} = \frac{1}{2}\rho {C_{\rm{d}}}(Rn)A(\alpha ){v_{\rm{r}}}\left| {{v_{\rm{r}}}} \right|$ (8)

 $Rn = \frac{{\rho \left| {{v_{\rm{c}}}} \right|d}}{\mu }$ (9)

 ${F_{{\rm{lift}}}} = \frac{1}{2}\rho v_{\rm{r}}^{\rm{2}}A(\alpha ){C_{\rm{l}}}(Rn, \alpha )$ (10)

 雷诺数 流态 Cd Cl Rn < 2·105 亚临界流 1 [3, 0.6] 2·105 < Rn < 5·105 临界流 [1, 0.4] 0.6 5·105 < Rn < 3·106 跨临界流 0.4 0.6

 ${\mathit{\boldsymbol{g}}^{\rm{l}}} = \left[ {\begin{array}{*{20}{c}} 0\\ 0\\ {9.81} \end{array}} \right]m/{s^2}$ (11)

 ${\mathit{\boldsymbol{f}}_{\rm{G}}}{\rm{(}}{\mathit{\boldsymbol{R}}_{\rm{t}}}) = - \mathit{\boldsymbol{R}}_{\rm{t}}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ W \end{array}} \right]$ (12)

 ${\mathit{\boldsymbol{f}}_{\rm{B}}}{\rm{(}}{\mathit{\boldsymbol{R}}_{\rm{t}}}{\rm{)}} = - \mathit{\boldsymbol{R}}_{\rm{t}}^{\rm{T}}\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ B \end{array}} \right]$ (13)

 ${\mathit{\boldsymbol{g}}_{{\rm{RB}}}}({\mathit{\boldsymbol{R}}_{\rm{t}}}{\rm{)}} = - \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{f}}_{\rm{G}}}({\mathit{\boldsymbol{R}}_{\rm{t}}}) + {\mathit{\boldsymbol{f}}_{\rm{B}}}({\mathit{\boldsymbol{R}}_{\rm{t}}})}\\ {\mathit{\boldsymbol{r}}_{\rm{C}}^{\rm{B}} \times {\mathit{\boldsymbol{f}}_{\rm{G}}}({\mathit{\boldsymbol{R}}_{\rm{t}}}) + \mathit{\boldsymbol{r}}_{\rm{B}}^{\rm{B}}{\mathit{\boldsymbol{f}}_{\rm{B}}}({\mathit{\boldsymbol{R}}_{\rm{t}}})} \end{array}} \right]$ (14)

 ${\mathit{\boldsymbol{g}}_{{\rm{RB}}}}(\mathit{\boldsymbol{q}})\left[ {\begin{array}{*{20}{c}} {(W - B)s\theta }\\ { - (W - B)c\theta s\phi }\\ { - (W - B)c\theta c\phi }\\ { - ({y_{\rm{G}}}W - {y_{\rm{B}}}B)c\theta c\phi + ({z_{\rm{G}}}W - {z_{\rm{B}}}B)c\theta s\phi }\\ {({z_{\rm{G}}}W - {z_{\rm{B}}}B)s\theta + ({x_{\rm{G}}}W - {x_{\rm{B}}}B)c\theta c\phi }\\ { - ({x_{\rm{G}}}W - {x_{\rm{B}}}B)c\theta s\phi - ({y_{\rm{G}}}W - {y_{\rm{B}}}B)s\theta } \end{array}} \right]$ (15)

 ${X_{\rm{A}}} = - {X_{\dot u}}\dot u$ (16)

 ${X_{\dot u}} = \frac{{\partial X}}{{\partial \dot u}}$ (17)

 ${\mathit{\boldsymbol{M}}_{\rm{A}}} = \mathit{\boldsymbol{M}}_{\rm{A}}^{\rm{T}} > 0$ (18)

 ${\mathit{\boldsymbol{C}}_{\rm{A}}}{\rm{(}}\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}) = - \mathit{\boldsymbol{C}}_{\rm{A}}^{\rm{T}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}), \forall q \in {\mathit{\boldsymbol{R}}^6}$ (19)

 ${\mathit{\boldsymbol{M}}_{\rm{A}}} = - {\rm{diag}}\left\{ {{X_{\dot u}}, {Y_{\dot v}}, {Z_{\dot w}}, {K_{\dot p}}, {M_{\dot q}}, {N_{\dot r}}} \right\}$ (20)
 ${\mathit{\boldsymbol{C}}_{\rm{A}}} = \left[ {\begin{array}{*{20}{c}} 0&0&0&0&{ - {Z_{\dot w}}w}&{{Y_{\dot v}}v}\\ 0&0&0&{{Z_{\dot w}}w}&0&{ - {X_{\dot u}}u}\\ 0&0&0&{ - {Y_{\dot v}}v}&{{X_{\dot u}}u}&0\\ 0&{ - {Z_{\dot w}}w}&{{Y_{\dot v}}v}&0&{ - {N_{\dot r}}r}&{{M_{\dot q}}q}\\ {{Z_{\dot w}}w}&0&{{X_{\dot u}}u}&{{N_{\dot r}}r}&0&{ - {K_{\dot p}}}\\ { - {Y_{\dot v}}v}&{{X_{\dot u}}u}&0&{ - {M_{\dot q}}q}&{{K_{\dot p}}p}&0 \end{array}} \right]$ (21)

 $\left\{ \begin{array}{l} {X_{\dot u}} = - 0.1m\\ {Y_{\dot v}} = - {\rm{ \mathit{ π} }}\rho {r^2}L\\ {Z_{\dot w}} = - {\rm{ \mathit{ π} }}\rho {r^2}L\\ {K_{\dot p}} = 0\\ {M_{\dot q}} = - \frac{1}{{12}}{\rm{ \mathit{ π} }}\rho {r^2}{L^3}\\ {N_{\dot r}} = - \frac{1}{{12}}{\rm{ \mathit{ π} }}\rho {r^2}L \end{array} \right.$ (22)

[20]中可以找到关于圆柱在流体中运动的附加质量效应的详细的理论和实验讨论, 它表明附加质量矩阵是状态相关的, 它的系数是圆柱运动距离的函数。

 $\mathit{\boldsymbol{D}}\left( {\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}} \right) > 0, \forall \mathit{\boldsymbol{q}} \in \mathit{\boldsymbol{R}}$ (23)

 $\begin{array}{l} \mathit{\boldsymbol{D}}(\mathit{\boldsymbol{q}}, \mathit{\boldsymbol{\dot q}}) = - {\rm{diag}}\left\{ {{X_{\dot u}}, {Y_{\dot v}}, {Z_{\dot w}}, {K_{\dot p}}, {M_{\dot q}}, {N_{\dot r}}} \right\} - \\ \;\;\;\;\;\;{\rm{diag}}\left\{ {{X_{u\left| u \right|}}\left| u \right|, {Y_{v\left| v \right|}}\left| v \right|, {Z_{w\left| w \right|}}\left| w \right|, {K_{p\left| p \right|}}\left| p \right|, {M_{q\left| q \right|}}\left| q \right|, {N_{r\left| r \right|}}\left| r \right|} \right\} \end{array}$ (24)

2 滑模自抗扰控制器

2.1 线性扩张观测器

 ${a_{11}}{{\ddot y}_1} + \cdots + {a_{1m}}{{\ddot y}_m} + {c_{11}}{{\dot y}_1} + \cdots + {c_{1m}}{{\dot y}_m} = {u_1} + {f_1}$ (25)

 $\begin{array}{l} {{\ddot y}_1} = - \frac{{{a_{12}}{{\ddot y}_2} + \cdots + {a_{1m}}{{\ddot y}_m} + {c_{11}}{{\dot y}_1} + \cdots + {c_{1m}}{{\dot y}_m} + {f_1}}}{{{a_{11}}}}\\ \;\; + \frac{1}{{{a_{11}}}}{u_1} \end{array}$ (26)

 $\left\{ \begin{array}{l} {\xi _1} = \frac{{{a_{12}}{{\ddot y}_2} + \cdots + {a_{1m}}{{\ddot y}_m} + {c_{11}}{{\dot y}_1} + \cdots + {c_{1m}}{{\dot y}_m} + {f_1}}}{{{a_{11}}}}\\ {b_1} = \frac{1}{{{a_{11}}}} \end{array} \right.$ (27)

 ${{\ddot y}_1} = {b_1}{u_1} + {\xi _1}$ (28)

 $\left\{ \begin{array}{l} {y_1} = {x_1}\\ {{\dot x}_1} = {x_2}\\ {{\dot x}_2} = {x_3} + {b_1}{u_1}\\ {{\dot x}_3} = w \end{array} \right.$ (29)

 $\left\{ \begin{array}{l} \dot x = Ax + Bu + Ef\\ y = Cx \end{array} \right.$ (30)

 $\left\{ \begin{array}{l} {e_1} = {z_1} - {y_1}\\ {{\dot z}_1} = {z_2} - {l_1}{e_1}\\ {{\dot z}_2} = {z_3} - {l_2}{e_1} + {b_1}{u_1}\\ {{\dot z}_3} = - {l_3}{e_1} \end{array} \right.$ (31)

 $[\begin{array}{*{20}{c}} {{l_1}}&{{l_2}}&{{l_3}} \end{array}] = [\begin{array}{*{20}{c}} {{\beta _1}{\omega _{{\rm{1o}}}}}&{{\beta _2}\omega _{1{\rm{o}}}^2}&{{\beta _3}\omega _{1{\rm{o}}}^2} \end{array}]$ (32)

 ${\beta _1} = 3{\omega _{1{\rm{o}}}}, {\beta _2} = 3\omega _{1{\rm{o}}}^2, {\beta _3} = \omega _{1{\rm{o}}}^3$ (33)

2.2 LESO收敛性及估计误差分析

 $\left\{ \begin{array}{l} {Z_1}(s) = \frac{{3{\omega _{1{\rm{o}}}}{s^2} + 3\omega _{1{\rm{o}}}^2s + \omega _{1o}^3}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{Y_1}(s) + \frac{{{b_1}s}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{U_1}(s)\\ {Z_2}(s) = \frac{{(3{\omega _{1{\rm{o}}}}{s^2} + \omega _{1{\rm{o}}}^3)s}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{Y_1}(s) + \frac{{{b_1}(s + 3{\omega _{1{\rm{o}}}})s}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{U_1}(s)\\ {Z_3}(s) = \frac{{\omega _{1{\rm{o}}}^3{s^2}}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{Y_1}(s) + \frac{{{b_1}\omega _{1{\rm{o}}}^3}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{U_1}(s) \end{array} \right.$ (34)

 $\left\{ \begin{array}{l} {e_1} = {z_1} - {y_1}\\ {e_2} = {z_2} - {{\dot y}_1}\\ {e_3} = {z_3} - {\xi _1} \end{array} \right.$ (35)

 ${\xi _1} = {x_3} = {{\dot x}_2} - {b_1}{u_1} = {{\ddot y}_1} - {b_1}{u_1}$ (36)

 $\left\{ \begin{array}{l} {E_1}(s) = \frac{{{s^3}}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{Y_1}(s) + \frac{s}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{b_1}{U_1}(s)\\ {E_2}(s) = - \frac{{(s + 3{\omega _{1{\rm{o}}}}){{\rm{s}}^3}}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{Y_1}(s) + \frac{s}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}{b_1}{U_1}(s)\\ {E_3}(s) = - \left[ {1 - \frac{{\omega _{1{\rm{o}}}^3}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}} \right]{s^3}{Y_1}(s) + \\ \;\;\;\;\;\;\;\;\;\;\;\left[ {1 - \frac{{\omega _{1{\rm{o}}}^3}}{{{{(s + {\omega _{1{\rm{o}}}})}^3}}}} \right]{b_1}{U_1}(s) \end{array} \right.$ (37)

 $\left\{ \begin{array}{l} {E_{{\rm{ss}}1}}(s) = \mathop {\lim }\limits_{s \to 0} s{E_1}(s) = 0\\ {E_{{\rm{ss2}}}}(s) = \mathop {\lim }\limits_{s \to 0} s{E_2}(s) = 0\\ {E_{{\rm{ss}}3}}(s) = \mathop {\lim }\limits_{s \to 0} s{E_3}(s) = 0 \end{array} \right.$ (38)

 $\dot e = (A - lC)e + E{\xi _1}$ (39)

 $\left\{ \begin{array}{l} {e_1} = \omega _{\rm{o}}^{ - 2}{\eta _1}\\ {e_2} = \omega _{\rm{o}}^{ - 1}{\eta _2}\\ {e_3} = {\eta _3} \end{array} \right.$ (40)

 $\mathit{\boldsymbol{e}} = \mathit{\boldsymbol{ \boldsymbol{\varLambda} }} \cdot \mathit{\boldsymbol{\eta }}$ (41)

 $\frac{1}{{{\omega _{\rm{o}}}}}\mathit{\boldsymbol{\dot \eta }}\mathit{ = }{\mathit{\boldsymbol{A}}_{\rm{o}}}\mathit{\boldsymbol{\eta }}\mathit{ + }\frac{1}{{{\omega _{\rm{o}}}}}E{\xi _1}$ (42)

 ${\mathit{\boldsymbol{A}}_{\rm{o}}} = \left[ {\begin{array}{*{20}{c}} { - {\beta _1}}&1&0\\ { - {\beta _2}}&0&1\\ { - {\beta _3}}&0&0 \end{array}} \right]$ (43)

 ${\lambda _i}\{ {\mathit{\boldsymbol{A}}_{\rm{o}}}\} = - 1{\rm{ < }}0,i = 1,2,3$ (44)

2.3 子系统的滑模变结构控制器

 $\dot V(t) \le - \alpha V(t) + \gamma (t), \forall t \ge {t_0} \ge 0$ (45)

 $V(t) \le {e^{ - \alpha (t - {t_0})}}V({t_0}) + \int_{{t_0}}^t {{e^{ - \alpha (t - \tau )}}} \gamma (\tau ){\rm{d}}\tau$ (46)

 ${{\ddot y}_1} = {b_1}u + {\xi _1}$ (47)

 $s = c\hat e + \dot {\hat e}$ (48)

 $u = \frac{{ - ks + {{\ddot y}_{{\rm{1d}}}} - c\dot {\hat e} - {{\hat \xi }_1}}}{{{b_1}}}$ (49)

 $V = \frac{1}{2}{s^2}$ (50)

 $\dot{s}=c\dot{\hat{e}}+\ddot{\hat{e}}=c\dot{\hat{e}}+{{{\ddot{\hat{y}}}}_{1}}-{{{\ddot{y}}}_{1d}}=c\dot{e}+{{b}_{1}}u+{{\xi }_{1}}-{{{\ddot{y}}}_{1d}}+{{{\hat{e}}}_{2}}$ (51)

 $\dot V = s\dot s = s\left( {c\dot {\hat e} + {b_1}{u_1} + {\xi _1} - {{\ddot y}_{{\rm{1d}}}} + {{\hat e}_2}} \right)$ (52)

 $\dot V = s(c\dot {\hat e} + {{\ddot y}_{{\rm{1d}}}} - {{\ddot y}_{{\rm{1d}}}} - k\hat s - c\dot {\hat e} + {\xi _1} - {{\hat \xi }_1} + {{\hat e}_2})$ (53)

 $\dot V = - k \cdot {s^2} + s({\xi _1} - {{\hat \xi }_1} + {{\hat e}_2})$ (54)

 $\dot V \le - k \cdot {s^2} + \frac{1}{2}{s^2} + \frac{1}{2}\Delta _{\max }^2 = - (2k - 1)V + \frac{1}{2}\Delta _{\max }^2$ (55)

 $\begin{array}{l} V(t) \le {e^{ - \alpha (t - {t_0})}}V({t_0}) + \int_{{t_0}}^t {{e^{ - \alpha (t - \tau )}}} \gamma (\tau ){\rm{d}}\tau = \\ {e^{ - (2k - 1)(t - {t_0})}}V({t_0}) + \frac{1}{2}\int_{{t_0}}^t {{e^{ - (2k - 1)(t - \tau )}}} {\rm{d}}\tau = \\ {e^{ - (2k - 1)(t - {t_0})}}V({t_0}) + \frac{1}{2}\int_{{t_0}}^t {{e^{ - (2k - 1)(t - \tau )}}} {\rm{d[}} - (2k - 1)(t - \tau {\rm{)] = }}\\ {e^{ - (2k - 1)(t - {t_0})}}V({t_0}) + \frac{1}{{2(2k - 1)}}\zeta _1^2(1 - {e^{ - (2k - 1)(t - {t_0})}}) \end{array}$ (56)

k > 1/2, 则有:

 $\mathop {\lim }\limits_{t \to \infty } V(t) \le \frac{1}{{2(2k - 1)}}\Delta _{\max }^2$ (57)

 $\mathop {\lim }\limits_{t \to \infty } V(t) = \frac{1}{{2(2k - 1)}}\Delta _{\max }^2$ (58)

 $\mathop {\lim }\limits_{t \to \infty } V(t) = 0$ (59)

2.4 微分跟踪器设计

 $\left\{ \begin{array}{l} {{\dot x}_1} = {x_2}\\ {{\dot x}_2} = - r{\mathop{\rm sgn}} \left( {{x_1} - {v_0}(t) + \frac{{{x_2}\left| {{x_2}} \right|}}{{2r}}} \right) \end{array} \right.$ (60)

3 仿真结果与分析

 $\left\{ {\begin{array}{*{20}{l}} {X = ({t^2} - 4.87t + 1)\sin (3t)}\\ {Y = ({t^2} - 4.87t + 1)\sin (4t)}\\ {Z = ({t^2} - 4.87t + 1)\sin (7t)} \end{array}} \right.$ (61)

 $\left\{ \begin{array}{l} {{\dot v}_{\rm{c}}} = - {\varepsilon _v}{v_{\rm{c}}} + {\omega _v}\\ {{\dot \alpha }_{\rm{c}}} = - {\varepsilon _\alpha }{\alpha _{\rm{c}}} + {\omega _\alpha }\\ {{\dot \beta }_{\rm{c}}} = - {\varepsilon _\beta }{\beta _{\rm{c}}} + {\omega _\beta } \end{array} \right.$ (62)

 图 4 UVMS位置变化轨迹 Fig. 4 Trajectory of UVMS's position

 图 5 UVMS位置跟踪误差 Fig. 5 Error in the UVMS's position tracking

 图 6 机械臂关节角度轨迹跟踪结果 Fig. 6 Angular trajectory tracking of manipulators

 图 7 UVMS机械臂期望关节角轨迹的跟踪误差 Fig. 7 Error in tracking the UVMS manipulator angular Trajectory

 图 8 不同控制器的控制输出对比 Fig. 8 Comparison of the control outputs of three controllers
4 结论

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