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Research on WMR Control of Visual Servo with Quantitative Feedback
DOI: 10.12677/AIRR.2022.111002, PDF, HTML, XML, 下载: 71  浏览: 151

Abstract: A quantitative feedback control method based on super-twisting algorithm is proposed aiming at the trajectory tracking of image visual servo mobile robot with external disturbance and quantitative feedback. Firstly, based on super-twisting sliding mode control plan, the disturbance term is put into the high-order derivative of sliding mode to reduce the chattering in sliding mode control; Secondly, the limit of quantization errors are put into sliding mode switching function to suppress the influence of quantization on system stability; it is proved by Lyapunov function that the system can converge to a small region related to quantization parameters in finite time. Finally, by dynamically adjusting quantization parameters, the system can finally converge to zero. The effectiveness of the control plan is verified by simulation results.

1. 引言

2. 视觉伺服WMR系统

IBVS系统如图1所示，WMR在天花板具有固定摄像机的全局坐标系下，p为相机捕捉的特征点,将WMR前进速度v与转向角速度w看作控制输入， $\left(x\left(t\right),y\left(t\right)\right)$ 为特征点p全局坐标。

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=v\mathrm{cos}\theta \left(t\right)-wd\mathrm{sin}\theta \left(t\right)\\ \stackrel{˙}{y}\left(t\right)=v\mathrm{sin}\theta \left(t\right)+wd\mathrm{cos}\theta \left(t\right)\\ \stackrel{˙}{\theta }\left(t\right)=w\end{array}$ (1)

IBVS系统为：

$\stackrel{˙}{P}\left(t\right)=\frac{1}{z\left(t\right)}D\left(t\right)u+d\left(t\right)$ (2)

$D\left(t\right)=\left[\begin{array}{cc}{h}_{11}-{x}_{p}\left(t\right){h}_{31}& {h}_{12}-{x}_{p}\left(t\right){h}_{32}\\ {h}_{21}-{y}_{p}\left(t\right){h}_{31}& {h}_{22}-{y}_{p}\left(t\right){h}_{32}\end{array}\right]\left[\begin{array}{cc}\mathrm{cos}\theta \left(t\right)& -d\mathrm{sin}\theta \left(t\right)\\ \mathrm{sin}\theta \left(t\right)& d\mathrm{cos}\theta \left(t\right)\end{array}\right]$ (3)

$z\left(t\right)$ 为深度， $D\left(t\right)$ 是可逆的 [3] $P\left(t\right)={\left({x}_{p}\left(t\right),{y}_{p}\left(t\right)\right)}^{\text{T}}$ 为图像系坐标， $d\left(t\right)={\left({d}_{x}\left(t\right),{d}_{y}\left(t\right)\right)}^{\text{T}}$ 为模型不确定、参数不确定和外界扰动的总和，WMR在图像系下的期望轨迹为 ${P}_{r}={\left({x}_{r},{y}_{r}\right)}^{\text{T}}$ 图像系下的误差定义为

$e\left(t\right)={P}_{r}-P\left(t\right)$ (4)

$s=e\left(t\right)+c{\int }_{0}^{t}e\left(t\right)\text{d}\tau$ (5)

$c>0$ 为待设定参数，对(5)求导可得

$\stackrel{˙}{s}=-\frac{1}{z\left(t\right)}D\left(t\right)u+ce\left(t\right)-d\left(t\right)$ (6)

$u=z\left(t\right)D{\left(t\right)}^{-1}\left\{ce\left(t\right)+{k}_{1}{|s|}^{\frac{1}{2}}sign\left(s\right)+{k}_{2}{\int }_{0}^{t}sign\left(s\right)\text{d}\tau \right\}$ (7)

(6)改写为

$\stackrel{˙}{s}=-{k}_{1}{|s|}^{\frac{1}{2}}sign\left(s\right)-{k}_{2}{\int }_{0}^{t}sign\left(s\right)\text{d}\tau +d\left(t\right)$ (8)

Figure 1. IBVS system description

3. 量化反馈STSMC

Figure 2. Quantitative feedback visual servo closed-loop system

$q\left(s\right)=\mu ×round\left(\frac{s}{\mu }\right)$ (9)

$\mu >0$ 为量化参数， $round\left(\text{ }\right)$ 表示就近取整运算，量化误差为

$q\left(s\right)-s={e}_{qs}$ (10)

$|{e}_{qs}|=|q\left(s\right)-s|\le \frac{\mu }{2}$ (11)

$\stackrel{˙}{s}=-{k}_{1}{|q\left(s\right)|}^{\frac{1}{2}}sign\left(q\left(s\right)\right)-{k}_{2}{\int }_{0}^{t}sign\left(q\left(s\right)\right)\text{d}\tau +d\left(t\right)$ (12)

${|q\left(s\right)|}^{\frac{1}{2}}={|{e}_{qs}+s|}^{\frac{1}{2}}\le {|{e}_{qs}|}^{\frac{1}{2}}+{|s|}^{\frac{1}{2}}\le {|s|}^{\frac{1}{2}}+\sqrt{\frac{\mu }{2}}$ (13)

$|s|\ge \frac{\mu }{2}$$sign\left(q\left(s\right)\right)=sign\left(s\right)$，当 $|s|\le \frac{\mu }{2}$$q\left(s\right)=0$，将(12)改写为

$\left\{\begin{array}{l}\stackrel{˙}{s}=-{k}_{1}{|s|}^{\frac{1}{2}}sign\left(s\right)+f\left({e}_{qs}\right)+r\\ \stackrel{˙}{r}=-{k}_{2}sign\left(s\right)+\stackrel{˙}{d}\left(t\right)\end{array}$ (14)

$\left\{\begin{array}{l}{k}_{1}>2\\ {k}_{2}>\frac{{k}_{1}^{3}+{\eta }^{2}\left(4{k}_{1}-8\right)}{{k}_{1}\left(4{k}_{1}-8\right)}\end{array}$ (15)

$\xi =\left[\begin{array}{c}{|s|}^{\frac{1}{2}}sign\left(s\right)\\ {k}_{2}{\int }_{0}^{t}sign\left(s\right)\text{d}\tau +d\left(t\right)\end{array}\right]$ (16)

$\Omega \left(\mu \right)=\frac{\sqrt{{\lambda }_{\mathrm{max}}\left(P\right)}}{\sqrt{{\lambda }_{\mathrm{max}}\left(P\right)}}\frac{{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}}{\left(1-\omega \right){\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}$ (17)

$\begin{array}{c}\stackrel{˙}{\xi }=\left[\begin{array}{c}\frac{1}{2|{\xi }_{1}|}\left(-{k}_{1}{|s|}^{\frac{1}{2}}sign\left(s\right)-{k}_{2}{\int }_{0}^{t}sign\left(s\right)\text{d}\tau +f\left({e}_{qs}\right)\right)\\ {k}_{2}sign\left(s\right)+\stackrel{˙}{d}\left(t\right)\end{array}\right]\\ =\frac{1}{|{\xi }_{1}|}\left[\begin{array}{cc}-\frac{{k}_{1}}{2}& \frac{1}{2}\\ -{k}_{2}+\delta \left(t\right)& 0\end{array}\right]\xi +\frac{1}{|{\xi }_{1}|}\left[\begin{array}{c}f\left({e}_{qs}\right)\\ 0\end{array}\right]\\ =\frac{1}{|{\xi }_{1}|}A\xi +\frac{1}{|{\xi }_{1}|}\left[\begin{array}{c}f\left({e}_{qs}\right)\\ 0\end{array}\right]\end{array}$ (18)

$V\left(\xi \right)={\xi }^{\text{T}}P\xi$ (19)

$P=\left[\begin{array}{cc}\frac{4{k}_{1}+{k}_{2}^{2}}{2}& \frac{-{k}_{2}}{2}\\ \frac{-{k}_{2}}{2}& 1\end{array}\right]$ (20)

$\begin{array}{c}\stackrel{˙}{V}\left(\xi \right)=\frac{1}{|{\xi }_{1}|}{\xi }^{\text{T}}\left({A}^{\text{T}}P+PA\right)\xi +\frac{1}{|{\xi }_{1}|}\left[f\left({e}_{qs}\right),0\right]P\xi \\ =-\frac{1}{|{\xi }_{1}|}{\xi }^{\text{T}}Q\left(\delta \left(t\right)\right)\xi +\frac{1}{|{\xi }_{1}|}f\left({e}_{qs}\right)\left[\begin{array}{cc}{p}_{11}& {p}_{12}\end{array}\right]\xi \end{array}$ (21)

$\stackrel{˙}{V}\left(\xi \right)\le -\frac{1}{|{\xi }_{1}|}{\xi }^{\text{T}}Q\left(\eta \right)\xi +\frac{1}{|{\xi }_{1}|}{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}‖\xi ‖$ (22)

$Q\left(\eta \right)=\left[\begin{array}{cc}{k}_{1}{k}_{2}+\frac{{k}_{2}^{3}}{2}-\frac{{k}_{2}^{2}}{4}-{\eta }^{2}& \frac{{k}_{2}}{2}-\frac{{k}_{2}^{2}}{2}\\ \frac{{k}_{2}}{2}-\frac{{k}_{2}^{2}}{2}& \frac{{k}_{2}}{2}-1\end{array}\right]$ (23)

${\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right){‖\xi ‖}^{2}\le {\xi }^{\text{T}}Q\left(\eta \right)\xi \le {\lambda }_{\mathrm{max}}\left(Q\left(\eta \right)\right){‖\xi ‖}^{2}$ (24)

$|{\xi }_{1}|\le ‖\xi ‖$$‖\xi ‖\le \sqrt{\frac{V\left(\xi \right)}{{\lambda }_{\mathrm{min}}\left(P\right)}}$ 由此可得

$\begin{array}{c}\stackrel{˙}{V}\left(\xi \right)\le -\frac{1}{|{\xi }_{1}|}{\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right){‖\xi ‖}^{2}+\frac{1}{|{\xi }_{1}|}{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}‖\xi ‖\\ \le -\frac{‖\xi ‖}{|{\xi }_{1}|}\left\{{\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)‖\xi ‖-{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}\right\}\\ \le -{\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)‖\xi ‖+{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}\\ \le -{\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)\left[\omega ‖\xi ‖-\left(1-\omega \right)‖\xi ‖\right]+{k}_{1}\sqrt{\frac{\mu }{2}}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}\end{array}$ (25)

$\forall ‖\xi ‖\ge {k}_{1}\sqrt{\frac{\mu }{2}}\frac{\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}}{\left(1-\omega \right){\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}$ 时有

$\stackrel{˙}{V}\left(\xi \right)\le \frac{-\omega {\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}{{\lambda }_{\mathrm{max}}^{\frac{1}{2}}\left(P\right)}{V}^{\frac{1}{2}}\left(\xi \right)$ (26)

${t}_{f}=\frac{2\sqrt{{\lambda }_{\mathrm{min}}\left(P\right)}\left\{{V}^{\frac{1}{2}}\left(0\right)-{k}_{1}\sqrt{\frac{\mu }{2}}\frac{\sqrt{{\lambda }_{\mathrm{max}}\left(P\right)}\sqrt{{p}_{11}^{2}+{p}_{12}^{2}}}{\left(1-\omega \right){\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}\right\}}{\omega {\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}$ (27)

$\stackrel{˙}{s}=-\frac{1}{z\left(t\right)}D\left(t\right)q\left(u\right)+ce\left(t\right)-d\left(t\right)$ (28)

$\left\{\begin{array}{l}\stackrel{˙}{s}=-{k}_{1}{|s|}^{\frac{1}{2}}sign\left(s\right)+f\left({e}_{qs}\right)+{e}_{qu}+r\\ \stackrel{˙}{r}=-{k}_{2}sign\left(s\right)+\stackrel{˙}{d}\left(t\right)\end{array}$ (29)

$‖\xi ‖\le {\Omega }^{\ast }\left(\mu ,{\mu }^{\ast }\right)=\frac{\sqrt{{\lambda }_{\mathrm{max}}\left(P\right)}}{\sqrt{{\lambda }_{\mathrm{max}}\left(P\right)}}\frac{\left({k}_{1}\sqrt{\frac{\mu }{2}+}\frac{{\mu }^{*}}{2}\right)\sqrt{{p}_{11}^{2}+{p}_{12}}}{\left(1-\omega \right){\lambda }_{\mathrm{min}}\left(Q\left(\eta \right)\right)}$ (30)

4. 数值与仿真

Figure 3. Dynamic uniform quantization of the system state

Figure 4. Dynamic uniform quantization of control inputs

Figure 5. Trajectory tracking error

Figure 6. Real trajectories

Figure 7. The state of the system under static quantization

Figure 8. Control input under static quantization

Figure 9. Systematic error under static quantization

5. 总结

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