Correction to: Configuration optimization of the feature-oriented reference system in large component assembly

Correction to: The International Journal of Advanced Manufacturing Technology

https://doi.org/10.1007/s00170-020-06554-6

The original article contained a mistake.

The symbol “L” and “K” should be replaced with ellipsis “⋯”.

Section 2.1:

$$ {{}^M\tau}_i^{ax}=\min \left\{{{}^{F_1}k}_i^{ax}{{}^{F_1}\tau}_i^{ax},{{}^{F_2}k}_i^{ax}{{}^{F_2}\tau}_i^{ax},\mathrm{L},{{}^{F_j}k}_i^{ax}{{}^{F_j}\tau}_i^{ax}\right\}\left( ax=x,y,z\right) $$

(2)

Section 2.3.2:

$$ {\boldsymbol{\Delta}}^T\mathbf{W}\boldsymbol{\Delta } \le {\boldsymbol{\Lambda}}^T\boldsymbol{\Lambda} $$

(19)

in which Δ =  diag (Δg, Δd^g, Δd^s, Δd^h, Δd^ef, Δd^mp, Δφ). $\boldsymbol{\Delta} \mathbf{g}=\mathit{\operatorname{diag}}\left(\Delta {\mathbf{g}}_1,\mathrm{L},\Delta {\mathbf{g}}_{n_1},\Delta {\mathbf{g}}_{n_1+1},\mathrm{L},\Delta {\mathbf{g}}_{n_1+{n}_2}\right)$,${\boldsymbol{\Delta} \mathbf{d}}^g=\mathit{\operatorname{diag}}\left(\Delta {d}_1^g,\mathrm{K},\Delta {d}_{n_3}^g\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^s=\mathit{\operatorname{diag}}\left(\Delta {d}_1^s,\mathrm{K},\Delta {d}_{n_4}^s\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^h=\mathit{\operatorname{diag}}\left(\Delta {d}_1^h,\mathrm{K},\Delta {d}_{n_5}^h\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^{ef}=\mathit{\operatorname{diag}}\left(\Delta {d}_1^{ef},\mathrm{K},\Delta {d}_{n_6}^{ef}\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^{mp}=\mathit{\operatorname{diag}}\left(\Delta {d}_1^{mp},\mathrm{K},\Delta {d}_{n_7}^{mp}\right)$, and $\boldsymbol{\Delta} \boldsymbol{\upvarphi} =\mathit{\operatorname{diag}}\left(\Delta {\varphi}_1,\mathrm{K},\Delta {\varphi}_{n_8}\right)$. W and Λ are in the forms of:

$$ \mathbf{W}=\mathit{\operatorname{diag}}\left({\mathbf{w}}_{{\mathbf{g}}_1},\mathrm{L},{\mathbf{w}}_{{\mathbf{g}}_{n_1}},{\mathbf{w}}_{{\mathbf{g}}_{n_1+1}},\mathrm{K},{\mathbf{w}}_{{\mathbf{g}}_{n_1+{n}_2}},{\mathbf{w}}_{\Delta {d}^g},{\mathbf{w}}_{\Delta {d}^s},{\mathbf{w}}_{\Delta {d}^h},{\mathbf{w}}_{\Delta {d}^{ef}},{\mathbf{w}}_{\Delta {d}^{mp}},{\mathbf{w}}_{\Delta \varphi}\right) $$

(20)

$$ \boldsymbol{\Lambda} =\mathit{\operatorname{diag}}\left({{}^M\boldsymbol{\Lambda}}_1,\mathrm{L},{{}^M\boldsymbol{\Lambda}}_{n_1},\frac{{{}^M\boldsymbol{\Lambda}}_{n_1+1}}{2},\mathrm{L},\frac{{{}^M\boldsymbol{\Lambda}}_{n_1+{n}_2}}{2},{\boldsymbol{\Lambda}}_{\Delta {d}^g},{\boldsymbol{\Lambda}}_{\Delta {d}^s},{\boldsymbol{\Lambda}}_{\Delta {d}^h},{\boldsymbol{\Lambda}}_{\Delta {d}^{ef}},{\boldsymbol{\Lambda}}_{\Delta {d}^{mp}},{\boldsymbol{\Lambda}}_{\Delta \varphi}\right) $$

(21)

where ${\boldsymbol{\Lambda}}_{\Delta {d}^g}=\mathit{\operatorname{diag}}\left(0.05,\mathrm{L},0.05\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^s}=\mathit{\operatorname{diag}}\left(0.05,\mathrm{L},0.05\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^h}=\mathit{\operatorname{diag}}\left(\frac{k_1^c{\tau}_1^c}{2},\mathrm{L},\frac{k_{n_5}^c{\tau}_{n_5}^c}{2}\right)$,${\boldsymbol{\Lambda}}_{\Delta {d}^{ef}}=\mathit{\operatorname{diag}}\left(\frac{k_1^{ef}{\tau}_1^{ef}}{2},\mathrm{L},\frac{k_{n_6}^{ef}{\tau}_{n_6}^{ef}}{2}\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^{mp}}=\mathit{\operatorname{diag}}\left(\frac{k_1^{mp}{\tau}_1^{mp}}{2},\mathrm{L},\frac{k_{n_7}^{mp}{\tau}_{n_7}^{mp}}{2}\right)$, and ${\boldsymbol{\Lambda}}_{\Delta \varphi }=\mathit{\operatorname{diag}}\left({k}_1^{\varphi }{\tau}_1^{\varphi },\mathrm{L},{k}_{n_8}^{\varphi }{\tau}_{n_8}^{\varphi}\right)$. w_i is equal to 0 or 1.

Section 2.4:

$$ \min f\left({\mathbf{g}}_1,{\mathbf{g}}_2,\mathrm{L},{\mathbf{g}}_N\right)=\sum \limits_{st=1}^{ST}\sum \limits_{i=1}^N{{}^{st}a}_i{\left\Vert {\mathbf{g}}_i-{{}^{st}{\mathbf{p}}^{\prime}}_i\right\Vert}^2 $$

(22)

Section 2.4:

$$ f\left({\mathbf{g}}_1,{\mathbf{g}}_2,\mathrm{L},{\mathbf{g}}_N\right)=\sum \limits_{i=1}^N{\left\Vert {\mathbf{g}}_i-{\mathbf{p}}_i^{{\prime\prime}}\right\Vert}^2+\beta \sum \limits_{i=1}^m\max \left(0,{\mathbf{Q}}_{ii}\right) $$

(23)

Section 3:

$$ {N}_1+{N}_2+\mathrm{L}+{N}_m\ge N\kern0.3em \left(3<{N}_{st}\le N\right) $$

(24)

Section 3:

where t_ax is the binary digits, and let ${b}_{t_{ax}}^{ax}\mathrm{L}{b}_3^{ax}{b}_2^{ax}{b}_1^{ax}\left( ax=x,y,z\right)$ represent the coordinate component. The conversion between them is:

$$ {g}^n={g}_{\mathrm{min}}^{ax}+\left({g}_{\mathrm{max}}^{ax}-{g}_{\mathrm{min}}^{ax}\right)\ast \sum \limits_{i=1}^{t_{ax}}{b}_i^{ax}\cdot {2}^{i-1}/\left({2}^{t_{ax}}-1\right) $$

(25)

The corresponding binary value of point g_j is ${\mathbf{b}}_j={\mathbf{b}}_j^x{\mathbf{b}}_j^y{\mathbf{b}}_j^z$, and the particle is x = [b₁, b₂, L, b_N].

Section 3.2:

$$ \mathbf{L}=\mathbf{S}-\mathbf{D} $$

(31)

where D =  diag (d₁, d₂, L, d_n).

Section 3.2:

$$ \mathbf{V}=\left[{\mathbf{v}}_1,{\mathbf{v}}_2,\mathrm{L},{\mathbf{v}}_{k_1}\right] $$

(32)

The correct equations should be the below:

Section 2.1:

$$ {{}^M\tau}_i^{ax}=\min \left\{{{}^{F_1}k}_i^{ax}{{}^{F_1}\tau}_i^{ax},{{}^{F_2}k}_i^{ax}{{}^{F_2}\tau}_i^{ax},\cdots, {{}^{F_j}k}_i^{ax}{{}^{F_j}\tau}_i^{ax}\right\}\left( ax=x,y,z\right) $$

(2)

Section 2.3.2:

$$ {\boldsymbol{\Delta}}^T\mathbf{W}\boldsymbol{\Delta } \le {\boldsymbol{\Lambda}}^T\boldsymbol{\Lambda} $$

(19)

in which Δ =  diag (Δg, Δd^g, Δd^s, Δd^h, Δd^ef, Δd^mp, Δφ). $\boldsymbol{\Delta} \mathbf{g}=\mathit{\operatorname{diag}}\left(\Delta {\mathbf{g}}_1,\cdots, \Delta {\mathbf{g}}_{n_1},\Delta {\mathbf{g}}_{n_1+1},\cdots, \Delta {\mathbf{g}}_{n_1+{n}_2}\right)$,${\boldsymbol{\Delta} \mathbf{d}}^g=\mathit{\operatorname{diag}}\left(\Delta {d}_1^g,\cdots, \Delta {d}_{n_3}^g\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^s=\mathit{\operatorname{diag}}\left(\Delta {d}_1^s,\cdots, \Delta {d}_{n_4}^s\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^h=\mathit{\operatorname{diag}}\left(\Delta {d}_1^h,\cdots, \Delta {d}_{n_5}^h\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^{ef}=\mathit{\operatorname{diag}}\left(\Delta {d}_1^{ef},\cdots, \Delta {d}_{n_6}^{ef}\right)$, ${\boldsymbol{\Delta} \mathbf{d}}^{mp}=\mathit{\operatorname{diag}}\left(\Delta {d}_1^{mp},\cdots, \Delta {d}_{n_7}^{mp}\right)$, and $\boldsymbol{\Delta} \boldsymbol{\upvarphi} =\mathit{\operatorname{diag}}\left(\Delta {\varphi}_1,\cdots, \Delta {\varphi}_{n_8}\right)$. W and Λ are in the forms of:

$$ \mathbf{W}=\mathit{\operatorname{diag}}\left({\mathbf{w}}_{{\mathbf{g}}_1},\cdots, {\mathbf{w}}_{{\mathbf{g}}_{n_1}},{\mathbf{w}}_{{\mathbf{g}}_{n_1+1}},\cdots, {\mathbf{w}}_{{\mathbf{g}}_{n_1+{n}_2}},{\mathbf{w}}_{\Delta {d}^g},{\mathbf{w}}_{\Delta {d}^s},{\mathbf{w}}_{\Delta {d}^h},{\mathbf{w}}_{\Delta {d}^{ef}},{\mathbf{w}}_{\Delta {d}^{mp}},{\mathbf{w}}_{\Delta \varphi}\right) $$

(20)

$$ \boldsymbol{\Lambda} =\mathit{\operatorname{diag}}\left({{}^M\boldsymbol{\Lambda}}_1,\cdots, {{}^M\boldsymbol{\Lambda}}_{n_1},\frac{{{}^M\boldsymbol{\Lambda}}_{n_1+1}}{2},\cdots, \frac{{{}^M\boldsymbol{\Lambda}}_{n_1+{n}_2}}{2},{\boldsymbol{\Lambda}}_{\Delta {d}^g},{\boldsymbol{\Lambda}}_{\Delta {d}^s},{\boldsymbol{\Lambda}}_{\Delta {d}^h},{\boldsymbol{\Lambda}}_{\Delta {d}^{ef}},{\boldsymbol{\Lambda}}_{\Delta {d}^{mp}},{\boldsymbol{\Lambda}}_{\Delta \varphi}\right) $$

(21)

where ${\boldsymbol{\Lambda}}_{\Delta {d}^g}=\mathit{\operatorname{diag}}\left(0.05,\cdots, 0.05\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^s}=\mathit{\operatorname{diag}}\left(0.05,\cdots, 0.05\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^h}=\mathit{\operatorname{diag}}\left(\frac{k_1^c{\tau}_1^c}{2},\cdots, \frac{k_{n_5}^c{\tau}_{n_5}^c}{2}\right)$,${\boldsymbol{\Lambda}}_{\Delta {d}^{ef}}=\mathit{\operatorname{diag}}\left(\frac{k_1^{ef}{\tau}_1^{ef}}{2},\cdots, \frac{k_{n_6}^{ef}{\tau}_{n_6}^{ef}}{2}\right)$, ${\boldsymbol{\Lambda}}_{\Delta {d}^{mp}}=\mathit{\operatorname{diag}}\left(\frac{k_1^{mp}{\tau}_1^{mp}}{2},\cdots, \frac{k_{n_7}^{mp}{\tau}_{n_7}^{mp}}{2}\right)$, and ${\boldsymbol{\Lambda}}_{\Delta \varphi }=\mathit{\operatorname{diag}}\left({k}_1^{\varphi }{\tau}_1^{\varphi },\cdots, {k}_{n_8}^{\varphi }{\tau}_{n_8}^{\varphi}\right)$. w_i is equal to 0 or 1.

Section 2.4:

$$ \min f\left({\mathbf{g}}_1,{\mathbf{g}}_2,\cdots, {\mathbf{g}}_N\right)=\sum \limits_{st=1}^{ST}\sum \limits_{i=1}^N{{}^{st}a}_i{\left\Vert {\mathbf{g}}_i-{{}^{st}{\mathbf{p}}^{\prime}}_i\right\Vert}^2 $$

(22)

Section 2.4:

$$ f\left({\mathbf{g}}_1,{\mathbf{g}}_2,\cdots, {\mathbf{g}}_N\right)=\sum \limits_{i=1}^N{\left\Vert {\mathbf{g}}_i-{\mathbf{p}}_i^{{\prime\prime}}\right\Vert}^2+\beta \sum \limits_{i=1}^m\max \left(0,{\mathbf{Q}}_{ii}\right) $$

(23)

Section 3:

$$ {N}_1+{N}_2+\cdots +{N}_m\ge N\kern0.3em \left(3<{N}_{st}\le N\right) $$

(24)

Section 3:

where t_ax is the binary digits, and let ${b}_{t_{ax}}^{ax}\cdots {b}_3^{ax}{b}_2^{ax}{b}_1^{ax}\left( ax=x,y,z\right)$ represent the coordinate component. The conversion between them is:

$$ {g}^n={g}_{\mathrm{min}}^{ax}+\left({g}_{\mathrm{max}}^{ax}-{g}_{\mathrm{min}}^{ax}\right)\ast \sum \limits_{i=1}^{t_{ax}}{b}_i^{ax}\cdot {2}^{i-1}/\left({2}^{t_{ax}}-1\right) $$

(25)

The corresponding binary value of point g_j is ${\mathbf{b}}_j={\mathbf{b}}_j^x{\mathbf{b}}_j^y{\mathbf{b}}_j^z$, and the particle is x = [b₁, b₂, ⋯, b_N].

Section 3.2:

$$ \mathbf{L}=\mathbf{S}-\mathbf{D} $$

(31)

where D =  diag (d₁, d₂, ⋯, d_n).

Section 3.2:

$$ \mathbf{V}=\left[{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots, {\mathbf{v}}_{k_1}\right] $$

(32)

The original article has been corrected.