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Grinding method of noncircular glass ornaments based on successive approximation and discrete cross-coupling iterative learning control

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Abstracts

In order to solve the problem of complicated calculation and large counter error of curve fitting method in industrial processing of noncircular glass ornaments, this paper proposes a grinding point searching algorithm. The successive approximation is used to find the grinding point of the workpiece contour and the grinding wheel. The proposed algorithm does not require complicated formula derivation and calculation and has good counter accuracy. According to the discrete characteristics of point-cloud data, a discrete cross-coupling iterative learning control algorithm is proposed to suppress the contour processing control error. Based on the norm theory, the convergence analysis and convergence conditions of the algorithm are presented. The effectiveness of the proposed algorithms is verified by experiments.

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Data availability

The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.

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Not applicable.

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Funding

This study has been supported by the National Key Research and Development Project of China (Grants No. 2016YFC1400302), the National Science Foundation of China (61871163), the Zhejiang Provincial Natural Science Foundation of China (No. LQ19E070003), and Zhejiang Provincial Key Lab of Equipment Electronics.

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Authors and Affiliations

  1. The College of Electronics and Information, Hangzhou Dianzi University, Hangzhou, 310018, China

    Xungen Li, Shuaishuai Lv, Mian Pan, Qi Ma, Wenyu Cai & Haibin Yu

  2. Pujiang Microelectronics and Intelligent Manufacturing Research Institute of Hangzhou Dianzi University, Jinhua, 322200, China

    Xungen Li & Shuaishuai Lv

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  1. Xungen Li
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  2. Shuaishuai Lv
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  3. Mian Pan
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  4. Qi Ma
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  5. Wenyu Cai
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Contributions

Xungen Li: conceptualization and methodology. Shuaishuai Lv: data curation, validation, and writing-original draft preparation. Mian Pan: visualization and investigation. Qi Ma: supervision. Wenyu Cai: writing-reviewing and editing. Haibin Yu: writing-reviewing and editing.

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Correspondence to Shuaishuai Lv.

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Appendix

Appendix

Proof: substitute Eq (8) into Eq. (11) to get

$$ {\displaystyle \begin{array}{c}\begin{array}{c}{u}_{k+1}(j)={u}_k(j)+{\varGamma}_d{e}_k\left(j+1\right)\\ {}+\left(\omega +\frac{1}{2\rho }{e}_k\left(j+1\right)\right){\overset{\sim }{\varGamma}}_{\varepsilon, d}\\ {}\times \left[{\omega}^T{e}_k\left(j+1\right)+\frac{1}{2\rho }{e^T}_k\left(j+1\right){e}_k\left(j+1\right)\right]\\ {}={u}_k(j)+{\varGamma}_d{e}_k\left(j+1\right)\\ {}+\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_k\left(j+1\right)+\frac{1}{2\rho }{e}_k\left(j+1\right)\\ {}\times {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_k\left(j+1\right)\\ {}+\frac{1}{2\rho}\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{e_k}^T\left(j+1\right){e}_k\left(j+1\right)\end{array}\\ {}+\frac{1}{4{\rho}^2}{e}_k\left(j+1\right){\overset{\sim }{\varGamma}}_{\varepsilon, d}{e^T}_k\left(j+1\right){e}_k\left(j+1\right)\end{array}} $$
(23)

Equation (15) can be rewritten as follows:

$$ {\displaystyle \begin{array}{c}{e}_k(j)={y}_d(j)-{y}_k(j)\\ {}={y}_d(j)-{y}_{k-1}(j)-{y}_k(j)+{y}_{k-1}(j)\\ {}={e}_{k-1}(j)-\left({y}_k(j)-{y}_{k-1}(j)\right)\\ {}={e}_{k-1}(j)-C\left({x}_k(j)-{x}_{k-1}(j)\right)\\ {}={e}_{k-1}(j)-C\Big(A{x}_k\left(j-1\right)+B{u}_k\left(j-1\right)\\ {}-A{x}_{k-1}\left(j-1\right)-B{u}_{k-1}\left(j-1\right)\Big)\\ {}={e}_{k-1}(j)- CA\left({x}_k\left(j-1\right)-{x}_{k-1}\left(j-1\right)\right)\\ {}- CB\left({u}_k\left(j-1\right)-{u}_{k-1}\left(j-1\right)\right)\\ {}={e}_{k-1}(j)- CB{\varGamma}_d{e}_{k-1}(j)\\ {}- CA\left({x}_k\left(j-1\right)-{x}_{k-1}\left(j-1\right)\right)\\ {}- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_{k-1}(j)\\ {}-\frac{1}{2\rho } CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_{k-1}(j)\\ {}-\frac{1}{2\rho } CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{e_{k-1}}^T(j){e}_{k-1}(j)\\ {}-\frac{1}{4{\rho}^2} CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{e^T}_{k-1}(j){e}_{k-1}(j)\end{array}} $$
(24)

Reorganize Eq. (24) to get

$$ {\displaystyle \begin{array}{c}\kern0.24em {e}_k(j)\\ {}=\left(I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right){e}_{k-1}(j)\\ {}- CA\left({x}_k\left(j-1\right)-{x}_{k-1}\left(j-1\right)\right)\\ {}-\frac{1}{2\rho } CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_{k-1}(j)\\ {}-\frac{1}{2\rho } CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{e_{k-1}}^T(j){e}_{k-1}(j)\\ {}-\frac{1}{4{\rho}^2} CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{e^T}_{k-1}(j){e}_{k-1}(j)\end{array}} $$
(25)

For any given control input uk(j),j ∈ [0, N], the general solution xk(j) to the traffic system

$$ {x}_k(j)={A}^j{x}_k(0)+\sum \limits_{i=0}^{j-1}{A}^{j-i-1}{Bu}_k(i) $$
(26)

can get

$$ {\displaystyle \begin{array}{c}{x}_k\left(j-1\right)-{x}_{k-1}\left(j-1\right)\\ {}=\sum \limits_{i=0}^{j-2}{A}^{j-i-2}B\left({u}_k\left(i-1\right)-{u}_{k-1}\left(i-1\right)\right)\\ {}\sum \limits_{i=0}^{j-2}{A}^{j-i-2}B\Big({\varGamma}_d{e}_{k-1}(i)\\ {}+\tau \left(i+1\right){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\varepsilon}_{k-1}(i)\Big)\\ {}\kern1em =\sum \limits_{i=0}^{j-2}{A}^{j-i-2}B{\varGamma}_d{e}_{k-1}(i)\\ {}+\sum \limits_{i=0}^{j-2}{A}^{j-i-2} B\tau (i){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\varepsilon}_{k-1}(i)\Big)\end{array}} $$
(27)
$$ {\displaystyle \begin{array}{c}{e}_k(j)=\left(I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right){e}_{k-1}(j)\\ {}- CA\sum \limits_{i=0}^{j-2}{A}^{j-i-2}B{\varGamma}_d{e}_{k-1}(i)\\ {}\kern0.84em - CA\sum \limits_{i=0}^{j-2}{A}^{j-i-2} B\tau (i){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\varepsilon}_{k-1}(i)\\ {}\kern0.96em -\frac{1}{2\rho } CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T{e}_{k-1}(j)\\ {}\kern0.96em -\frac{1}{2\rho } CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{e_{k-1}}^T(j){e}_{k-1}(j)\\ {}\kern0.84em -\frac{1}{4{\rho}^2} CB{e}_{k-1}(j){\overset{\sim }{\varGamma}}_{\varepsilon, d}{e^T}_{k-1}(j){e}_{k-1}(j)\end{array}} $$
(28)

Take the norm on both ends of (28) to get

$$ {\displaystyle \begin{array}{c}\begin{array}{c}\kern0.36em \left\Vert {e}_k(j)\right\Vert \\ {}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\left\Vert CA\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert {A}^{j-i-2}\right\Vert \left\Vert B{\varGamma}_d\right\Vert \left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\left\Vert CA\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert {A}^{j-i-2}\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert \tau (i)\right\Vert \left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e_{k-1}}^T(j){e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e^T}_{k-1}(j){e}_{k-1}(j)\right\Vert \end{array}\\ {}\begin{array}{c}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert \tau (i)\right\Vert \left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e_{k-1}}^T(j){e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e^T}_{k-1}(j){e}_{k-1}(j)\right\Vert \end{array}\end{array}} $$
(29)

Define κ ≔ max {‖Aj − i − 2‖, j ∈ [0, N], i ∈ [0, j − 2]}, Eq. (29) can be sorted out.

$$ {\displaystyle \begin{array}{c}\begin{array}{c}\kern0.36em \left\Vert {e}_k(j)\right\Vert \\ {}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert \tau (i)\right\Vert \left\Vert {e}_{k-1}(i)\right\Vert \end{array}\\ {}\begin{array}{c}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e_{k-1}}^T(j)\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e_{k-1}}^T(j)\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \sum \limits_{i=0}^{j-2}\left\Vert \tau (i)\right\Vert \left\Vert {e}_{k-1}(i)\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert CB\right\Vert \left\Vert {e}_{k-1}(j)\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \end{array}\end{array}} $$
(30)

By multiplying both sides of (30) by a−λj and making use of some trivial manipulations, we can obtain that for any j ∈ [0, N], what we can obtain after finishing is

$$ {\displaystyle \begin{array}{c}\begin{array}{c}\;{\left\Vert {e}_k(j)\right\Vert}_{\lambda}\\ {}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert {\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert {u}_k\sum \limits_{i=0}^{j-2}{a}^{-\lambda \left(j-i\right)}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert {v}_k\sum \limits_{i=0}^{j-2}{a}^{-2\lambda \left(j-i\right)}\\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert {\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}+\frac{1}{2\rho}\left\Vert CB\right\Vert \left\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert {\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}+\frac{1}{4{\rho}^2}\left\Vert CB\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert {\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}\le \left\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert {\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert {u}_k\frac{a^{-\lambda}\left(1-{a}^{-\lambda \left(j-1\right)}\right)}{1-{a}^{-\lambda }}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert {v}_k\frac{a^{-2\lambda}\left(1-{a}^{-2\lambda \left(j-1\right)}\right)}{1-{a}^{-2\lambda }}\\ {}+\left\Vert CB\right\Vert \left(\frac{1}{2\rho}\right\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\left\Vert +\frac{1}{2\rho}\right\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\Big\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \Big){\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\end{array}\\ {}\begin{array}{c}=\left(\right\Vert I- CB{\varGamma}_d- CB\omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\Big\Vert \\ {}+\left\Vert CB\right\Vert \left(\frac{1}{2\rho}\right\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}{\omega}^T\left\Vert +\frac{1}{2\rho}\right\Vert \omega {\overset{\sim }{\varGamma}}_{\varepsilon, d}\Big\Vert \\ {}+\frac{1}{4{\rho}^2}\left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert \left)\right){\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B{\varGamma}_d\right\Vert {u}_k\frac{a^{-\lambda}\left(1-{a}^{-\lambda \left(j-1\right)}\right)}{1-{a}^{-\lambda }}\\ {}+\kappa \left\Vert CA\right\Vert \left\Vert B\right\Vert \left\Vert {\overset{\sim }{\varGamma}}_{\varepsilon, d}\right\Vert {v}_k\frac{a^{-2\lambda}\left(1-{a}^{-2\lambda \left(j-1\right)}\right)}{1-{a}^{-2\lambda }}\end{array}\end{array}} $$
(31)

where uk = max {‖ek − 1(i)‖λ| i ∈ [0, j − 2]},vk = max {‖τ(i)‖λ‖ek − 1(i)‖λ| i ∈ [0, j − 2]}.

By virtue of the contractive condition of (22),

\( {\displaystyle \begin{array}{l}\left\Vert \boldsymbol{I}-\boldsymbol{CB}{\varGamma}_d-\boldsymbol{CB}\omega {\tilde{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \\ {}+\left\Vert \boldsymbol{CB}\right\Vert \left(\frac{1}{2\rho}\left\Vert {\tilde{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert +\frac{1}{2\rho}\left\Vert \omega {\tilde{\varGamma}}_{\varepsilon, d}\right\Vert +\frac{1}{4{\rho}^2}\left\Vert {\tilde{\varGamma}}_{\varepsilon, d}\right\Vert \right)<1\end{array}} \) with a sufficiently large λ, we can obtain

$$ {\displaystyle \begin{array}{l}{\left\Vert {\boldsymbol{e}}_k(j)\right\Vert}_{\lambda}\le \Big(\left\Vert \boldsymbol{I}-\boldsymbol{CB}{\varGamma}_d-\boldsymbol{CB}\omega {\tilde{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert \\ {}+\left\Vert \boldsymbol{CB}\right\Vert \left(\frac{1}{2\rho}\left\Vert {\tilde{\varGamma}}_{\varepsilon, d}{\omega}^T\right\Vert +\frac{1}{2\rho}\left\Vert \omega {\tilde{\varGamma}}_{\varepsilon, d}\right\Vert +\frac{1}{4{\rho}^2}\left\Vert {\tilde{\varGamma}}_{\varepsilon, d}\right\Vert \right)\Big){\left\Vert {e}_{k-1}(j)\right\Vert}_{\lambda}\end{array}} $$
(32)

Taking limit as k → ∞, we have \( \underset{k\to \infty }{\lim }{\left\Vert {\boldsymbol{e}}_k(j)\right\Vert}_{\lambda }=0 \), and based on Eq. (18), we can obtain \( \underset{k\to \infty }{\lim }{\left\Vert {\varepsilon}_k(j)\right\Vert}_{\lambda }=0 \).

That is, as the number of iterations increases, the single-axis error and contour error both tend to 0.

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Li, X., Lv, S., Pan, M. et al. Grinding method of noncircular glass ornaments based on successive approximation and discrete cross-coupling iterative learning control. Int J Adv Manuf Technol 114, 3823–3835 (2021). https://doi.org/10.1007/s00170-021-07076-5

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