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 自动化学报  2017, Vol. 43 Issue (4): 538-547 PDF

1. 西北工业大学航天学院智能机器人研究中心 西安 710072;
2. 西北工业大学航天飞行动力学技术国家重点实验室 西安 710072

Capturing the Target for a Tethered Space Robot Using Robust Adaptive Controller
HUANG Pan-Feng1,2, HU Yong-Xin1,2, WANG Dong-Ke1,2, MENG Zhong-Jie1,2, LIU Zheng-Xiong1,2
1. Research Center for Intelligent Robotics, School of Astronautics, Northwestern Polytechnical University, Xi'an 710072;
2. National Key Laboratory of Aerospace Flight Dynamics, Northwestern Polytechnical University, Xi'an 710072
Foundation Item: Supported by National Natural Science Foundation of China (11272256, 60805034, 61005062)
Corresponding author. HUANG Pan-Feng Professor at the School of Astronautics, Northwestern Polytechnical University. His research interest covers space robotics, tethered space robotics, intelligent control, machine vision, and space teleoperation. Corresponding author of this paper
Recommended by Associate Editor SUN Fu-Chun
Abstract: Aimed at the problem of stabilization for a tethered space robot (TSR) capturing a target, a position-based impedance control is presented based on the model of a tethered space robot system. For the problem of model uncertainty of the tethered space robot system, an artificial neutral network (ANN) is used to estimate and compensate for the uncertainty, and a robust term is designed to repress the interference of tether and the effect of the estimation deviation by the ANN. Then a robust and adaptive controller for the TSR capturing a target is designed, and the stabilization of the controller is demonstrated. For the purpose of comparison, a simulation for an adaptive controller without the robust term is made, and the result shows that the controller designed in this paper can guarantee the stabilization during the TSR capturing a target. Compared to the adaptive controller without the robust term, the robust adaptive controller can compensate for the uncertainty effectively, with smaller overshoot, less convergence time, and higher control accuracy during the control process.
Key words: Tethered space robot (TSR)     target capturing     impedance control     robust adaptive control

1 动力学模型建立

 图 1 空间绳系机器人目标抓捕示意图 Figure 1 Target capture of the TSR

1.1 基本假设

$i$段空间系绳在空间平台轨道坐标系下的矢量为${\pmb l}_i$, 则

 ${\pmb l}_i=[{l_i}\cos{{\alpha}_i}\cos{{\beta}_i},\,{l_i}\sin{{\alpha}_i}\cos{{\beta}_i},\,{l_i}\sin{{\beta}_i}]^{\rm T}$ (1)

$i (i\le n-1)$个珠点在空间平台轨道坐标系下的位置可以表示为

 ${\pmb r}_i=\sum_{j=1}^i {{\pmb l}_j}$ (2)

$i (i\le n-1)$个珠点的速度可以表示为

 ${\pmb V}_i={{\pmb V}_0}+\dot{{\pmb r}}_i+ {{\pmb \omega}} \times {{\pmb r}_i}$ (3)

 ${\pmb r}_b={{\pmb r}_n}+ {{\pmb d}}$ (4)

 ${\pmb R}_b={{\pmb R}_0}+ {{\pmb r}_b}$ (5)

 ${\pmb V}_b={\pmb V}_0+\dot{{\pmb r}}_b+ {{\pmb \omega}} \times {{\pmb r}_b}$ (6)

 $\dot{{\pmb r}}_b=\dot{{\pmb r}}_n+ \dot{{\pmb d}}=\sum_{i=1}^n {\dot{{\pmb l}}_i}+\dot{{\pmb d}}$ (7)

 ${\pmb \omega}_b=\ [\dot{\theta}\sin{{\psi}}+\dot{\varphi}\cos{{\theta}\cos{\psi}}, \dot{\theta}\cos{\psi}\,-\nonumber\\ \;\;\;\;\;\; \dot{\varphi}\cos{{\theta}}\sin{\psi},\dot{\varphi}\sin{\theta}+\dot{\psi}]^{\rm T}$ (8)

1.2 系统动能

1) 空间平台动能为

 ${T}_p=\frac{1}{2}{{{\pmb V}}_0}^{\rm T}{M}{{{\pmb V}}_0}$ (9)

2) 第$i$个珠点的动能为

 ${T}_i=\frac{1}{2}{{{\pmb V}}_i}^{\rm T}{m}_i{{{\pmb V}}_i}$

 ${T}_l=\sum_{i=1}^{n-1}{{{\pmb T}}_i}$ (10)

3) 空间绳系机器人线动能为

 ${T}_r=\frac{1}{2}{{{\pmb V}}_b}^{\rm T}{m}_n{{{\pmb V}}_b}$ (11)

4) 空间绳系机器人转动动能为

 ${T}_a=\frac{1}{2}{{{\pmb \omega}}_b}^{\rm T}{{\pmb I}}{{{\pmb \omega}}_b}$ (12)

1) 空间绳系机器人系统广义惯量矩阵${M}$是正定矩阵;

2) 矩阵$\dot{{ M}}-2{ N}$为斜对称矩阵, 且具有以下性质:

 ${{\pmb \chi}}^{\rm T}(\dot{{ M}}-2{ N}) {{\pmb \chi}}=0$ (21)

 ${ M}_d\ddot{{\pmb e}}_i+{ B}_d\dot{{\pmb e}}_i+{ K}_d{\pmb e}_i=-{ \pmb F}_{ei}$ (22)
 图 2 空间绳系机器人碰撞力示意图 Figure 2 The collision force of the TSR

 ${\pmb x}_d={\pmb x}_r+\sum_{i=1}^7{\pmb e}_i$ (23)

 图 3 基于位置的空间绳系机器人阻抗控制 Figure 3 Block diagram of the impedance controller based on position for the TSR

 $({ M}_0+\Delta{ M})\ddot{{\pmb \xi}}+ ({ N}_0+\Delta{ N})\dot{{\pmb \xi}}\,+\notag\\ \;\;\;\;\;\; \qquad ({\pmb G}_0+\Delta{\pmb G})={\pmb Q}+{\pmb \tau}$ (24)

 $\begin{cases} { M}={ M}_0+\Delta{{ M}}\\ { N}={ N}_0+\Delta{{ N}}\\ {\pmb G}={\pmb G}_0+\Delta{{\pmb G}} \end{cases}$ (25)

 ${\pmb e}={\pmb \xi}_d-{\pmb \xi}$ (26)

 ${\pmb r}=\dot{{\pmb e}}+{ \Lambda}{\pmb e}=\dot{{\pmb \xi}}_d-\dot{{\pmb \xi}}+{ \Lambda}{\pmb e}$ (27)

 ${ M}_0\dot{{\pmb r}}= { M}_0(\ddot{{\pmb e}}+ { \Lambda}\dot{{\pmb e}})={ M}_0(\ddot{{\pmb \xi}}_d- \ddot{{\pmb \xi}}+{ \Lambda}\dot{{\pmb e}})=\notag\\ \;\;\;\;\;\; { M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})+\Delta{ M}\ddot{{\pmb \xi}}\,+ \notag\\ \;\;\;\;\;\; { N}\dot{{\pmb \xi}}+ {\pmb G}-{\pmb Q}-{\pmb \tau}=\notag\\ \;\;\;\;\;\; { M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})+ { N}_0(\dot{{\pmb \xi}}_d+{ \Lambda}{{\pmb e}})\,-\notag\\ \;\;\;\;\;\; { N}_0(\dot{{\pmb \xi}}_d+{ \Lambda}{{\pmb e}}- \dot{{\pmb \xi}})+{\pmb G}_0+\Delta{ M}\ddot{{\pmb \xi}}\,+\notag\\ \;\;\;\;\;\; \Delta{ N}\dot{{\pmb \xi}}+\Delta{\pmb G}-{\pmb Q}-{\pmb \tau}=\notag\\ \;\;\;\;\;\; { M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})+ { N}_0(\dot{{\pmb \xi}}_d+{ \Lambda}{{\pmb e}})\,-\notag\\ \;\;\;\;\;\; { N}_0{\pmb r} +{\pmb G}_0+{\pmb \rho}-{\pmb Q}-{\pmb \tau}$ (28)

 $M_0\dot{{\pmb r}}+{ N}_0{\pmb r} ={ M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})\,+\notag\\ \;\;\;\;\;\; \qquad { N}_0(\dot{{\pmb \xi}}_d+ { \Lambda}{{\pmb e}})+{\pmb G}_0+{\pmb \rho}-{\pmb Q}-{\pmb \tau}$ (29)

 ${\pmb Q}= { K}{\pmb r}+{ M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})\,+ \nonumber\\ \;\;\;\;\;\; { N}_0(\dot{{\pmb \xi}}_d+ { \Lambda}{{\pmb e}})+{\pmb G}_0+\hat{{\pmb \rho}}+{\pmb \eta}$ (30)

 $\begin{split} { M}_0\dot{{\pmb r}}=-{ N}_0{{\pmb r}}-{ K}{{\pmb r}}+\tilde{{\pmb \rho}}+{\pmb \delta}-{\pmb \eta}-{\pmb \tau} \end{split}$ (31)

 $\hat{{\pmb \rho}}={\hat{{\pmb \Theta}}}^{\rm T}{\pmb \Phi}_{\rho}$ (32)

 ${\pmb \Phi}_{{\rho}i}({\pmb Y}){\rm exp}\Bigg(-\frac{{\|{\pmb Y}-C_i\|}^2}{2{{\sigma}_i}^2}\Bigg)$ (33)

 $\dot{\hat{{\pmb \Theta}}}={ \pmb F}_{\rho}{{\pmb \Phi}_{\rho}}{\pmb r}^{\rm T}-k_{\rho}{ \pmb F}_{\rho}{\|{\pmb r}\|}\hat{{\pmb \Theta}}$ (34)

 ${\pmb \eta}=({\eta}_{\tilde{\rho}}+{\eta}_{\tau}){\rm sgn}({\pmb r})$ (35)

 ${\rm sat}({r}_i,\varepsilon)= \begin{cases} 1, &{r}_i>\varepsilon \\[1mm] \displaystyle\frac{{r}_i}{\varepsilon}, &-\varepsilon < {r}_i\le\varepsilon\\[1mm] -1, &{r}_i\le-\varepsilon \end{cases}$ (36)

3 稳定性证明

 $V=\frac{1}{2}{{\pmb r}^{\rm T}}{{ M}_0}{{\pmb r}}+\frac{1}{2}{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}{\tilde{{\pmb \Theta}}}\right)$ (40)

 $\dot{V}={{\pmb r}^{\rm T}}{{ M}_0}\dot{{\pmb r}}+\frac{1}{2}{{\pmb r}^{\rm T}}\dot{{ M}_0}{{\pmb r}}+{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)$ (41)

 $\dot{V}= {{\pmb r}^{\rm T}}{{ M}_0}\dot{{\pmb r}}+\frac{1}{2}{{\pmb r}^{\rm T}}\dot{{ M}_0}{{\pmb r}}+{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)=\notag\\ \;\;\;\;\;\; {{\pmb r}^{\rm T}}\Big\{-{ N}_0{\pmb r}+{ M}_0(\ddot{{\pmb \xi}}_d+{ \Lambda}\dot{{\pmb e}})+{ N}_0\left(\dot{{\pmb \xi}}_d+{ \Lambda}{{\pmb e}}\right)+ \notag\\ \;\;\;\;\;\; {\pmb G}_0+\!{\pmb \rho}-{\pmb Q}-{\pmb \tau}\Big\}\!+\! \frac{1}{2}{{\pmb r}^{\rm T}}\dot{{ M}_0}{{\pmb r}}\!+\!{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)\!\!=\notag\\ \;\;\;\;\;\; {\pmb r}^{\rm T}(-{ N}_0{{\pmb r}}-{ K}{{\pmb r}}+\tilde{{\pmb \rho}}+{\pmb \delta}-{\pmb \eta}-{\pmb \tau})\,+\notag\\ \;\;\;\;\;\; \frac{1}{2}{{\pmb r}^{\rm T}}\dot{{ M}_0}{{\pmb r}}+{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)=\notag\\ \;\;\;\;\;\; {\pmb r}^{\rm T}(-{ K}{{\pmb r}}+\tilde{{\pmb \rho}}+{\pmb \delta}-{\pmb \eta}-{\pmb \tau})+\notag\\ \;\;\;\;\;\; \frac{1}{2}{{\pmb r}^{\rm T}}(\dot{{ M}_0}-2{ N}_0){{\pmb r}}+{\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)$ (42)

 $\dot{V}= {\pmb r}^{\rm T}(-{ K}{{\pmb r}}+\tilde{\boldsymbol {\rho}}+{\pmb \delta}-{\pmb \eta}-{\pmb \tau})\,+\nonumber\\ \;\;\;\;\;\; {\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\tilde{{\pmb \Theta}}}\right)$ (43)

 $\dot{V}= {\pmb r}^{\rm T}(-{ K}{{\pmb r}}+\tilde{{\pmb \rho}}+ {\pmb \delta}-{\pmb \eta}-{\pmb \tau})\,+\nonumber\\ \;\;\;\;\;\; {\rm tr}\left({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\hat{{\pmb \Theta}}}\right)$ (44)

 $\dot{V}= -{{\pmb r}^{\rm T}}{Kr}+{{\pmb r}^{\rm T}}( \tilde{{\pmb \rho }} -{\pmb \eta} +{\pmb \delta} -{\pmb \tau})\,-\nonumber\\ \;\;\;\;\;\; {\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}{{{ \pmb F}}_{\rho }}^{-1} ( {{ \pmb F}_{\rho }}{{\pmb \Phi }_{\rho }}{{\pmb r}^{\rm T}}-{{k}_{\rho }}{{ \pmb F}_ {\rho }}\| {\pmb r} \|\hat{{\pmb \Theta }} ) \right) =\nonumber\\ \;\;\;\;\;\;-{{\pmb r}^{\rm T}}{Kr}+{{\pmb r}^{\rm T}}( \tilde{{\pmb \rho }}-{\pmb \eta} + {\pmb \delta} -{\pmb \tau})\,-\nonumber\\ \;\;\;\;\;\; {\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}({{\pmb \Phi }_{\rho }}{{\pmb r}^{\rm T}}-{{k}_{\rho }}\| {\pmb r} \|\hat{{\pmb \Theta }})\right) =\nonumber\\ \;\;\;\;\;\;-{{\pmb r}^{\rm T}}{Kr}+{{\pmb r}^{\rm T}}( \tilde{{\pmb \rho }}-{\pmb \eta} + {\pmb \delta} -{\pmb \tau})\,-\nonumber\\ \;\;\;\;\;\; {\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}{{\pmb \Phi }_{\rho }}{{\pmb r}^{\rm T}}\right)- {\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}(-{{k}_{\rho }}\| {\pmb r} \|\hat{{\pmb \Theta }})\right)=\nonumber\\ \;\;\;\;\;\;-{{\pmb r}^{\rm T}}{Kr}+{{\pmb r}^{\rm T}}( \tilde{{\pmb \rho }}-{\pmb \eta} + {\pmb \delta} -{\pmb \tau})\,-\nonumber\\ \;\;\;\;\;\; {{\pmb r}^{\rm T}}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}{{\pmb \Phi }_{\rho }}\right)+{\rm tr} \left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}({{k}_{\rho }}\| {\pmb r} \|\hat{{\pmb \Theta }})\right)=\nonumber\\ \;\;\;\;\;\;-{{\pmb r}^{\rm T}}{Kr}+{{\pmb r}^{\rm T}}(-{\pmb \eta} +{\pmb \delta} -{\pmb \tau})\,+\nonumber\\ \;\;\;\;\;\; {{{k}_{\rho }}\| {\pmb r} \|}{\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}} ({\pmb \Theta }-\tilde{{\pmb \Theta } })\right)$ (45)

${\rm tr}({\tilde{{\pmb \Theta}}^{\rm T}}{{ \pmb F}_{\rho}^{-1}}\dot{\hat{{\pmb \Theta}}})$满足以下性质:

 ${\rm tr}\left( {{{\tilde{{\pmb \Theta } }}}^{\rm T}}( {{\pmb \Theta } } -\tilde{{\pmb \Theta } } ) \right) ={{( \tilde{{\pmb \Theta } },{{\pmb \Theta } } )}_{\pmb F}}-\| {\tilde{{\pmb \Theta } }} \|_{\pmb F}^{2}\le\nonumber\\ \;\;\;\;\;\; \qquad \| {\tilde{{\pmb \Theta } }} \|_{\pmb F}^{{}} \|{{\pmb \Theta } } \|_{\pmb F}^{{}} -\| {\tilde{{\pmb \Theta } }} \|_{\pmb F}^{2}\le\nonumber\\ \;\;\;\;\;\; \qquad \| {\tilde{{\pmb \Theta } }} \|_{\pmb F}^{{}}\left( {{{\pmb \Theta } }_{\max }}-\| {\tilde{{\pmb \Theta } }} \|_{\pmb F}^{{}} \right)$ (46)

 ${{\pmb r}^{\rm T}}( -{\pmb \eta} +{\pmb \delta} -{\pmb \tau})=\nonumber\\ \;\;\;\;\;\; \qquad {{\pmb r}^{\rm T}}( -( {{\eta }_{\delta }}+{{\eta }_{\tau }} ) {\rm sgn}({\pmb r})+ {\pmb \delta} -{\pmb \tau} )\le 0$ (47)

 $\dot{V}\le -{{ K}_{\max }}{{\| {\pmb r} \|}^{2}}+{{k}_{\rho }}\| {\pmb r} \| \| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}}( {{\pmb \Theta }_{\max }} -\| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}} )$ (48)

$\dot{V}\le0$, 需要满足

 ${{ K}_{\max }}\| {\pmb r} \|\ge {{k}_{\rho }} \| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}}( {{{\pmb \Theta} }_{\max }}-\| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}} )$

 $\| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}}( {{\pmb \Theta }_{\max }}-\| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}} ) =\notag\\ \;\;\;\;\;\; \qquad -{{( \| {\tilde{{\pmb \Theta} }} \|_{\pmb F}^{{}}-0.5{{\pmb \Theta }_{\max }} )}^{2}}\,+\notag\\ \;\;\;\;\;\; \qquad 0.25{{\pmb \Theta }_{\max }}^{2} \le 0.25{{\pmb \Theta }_{\max }}^{2}$ (49)

 图 5 姿态角跟踪误差变化曲线 Figure 5 The tracking deviation of the attitude angle
 图 6 面内角跟踪误差变化曲线 Figure 6 The tracking deviation of the in-plane angle
 图 7 系绳长度跟踪误差变化曲线 Figure 7 The tracking deviation of the tether length
 图 8 抓捕目标欧拉姿态角变化曲线 Figure 8 The Eular attitude angle of the target
 图 9 抓捕目标位置变化曲线 Figure 9 The position of the target

5 结论

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