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A G3 continuous five-axis tool path corner smoothing method with improved machining efficiency and accurately controlled deviation of tool axis orientation

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Abstract

Parts with free-form surfaces are machined by connecting short linear tool path segments on five-axis computer numerical control (CNC) machining centers. Discontinuities at the junctions of linear segments reduce the machining efficiency, leave marks on the finish surfaces, and cause higher tracking errors, hence leading to larger contouring errors. This paper aims to improve the machining efficiency and accuracy of previously presented G3 continuous tool path smoothing method for 5-axis CNC machining (Sun and Altintas in CIRP J Manuf Sci Tec 32:529–549, [8]). Compared with the previous work, this paper improves the machining efficiency by optimizing the maximum feedrate along the transition splines while respecting the corner transition error tolerance, chord error tolerance, and tangential kinematic limits (i.e., the feedrate command, tangential acceleration, and jerk limits) at the connection points or “corners”. To improve the machining accuracy, this paper takes the maximum feedrate allowed on the tool tip position spline as one of the constraints to keep the path deviation of tool axis orientation within the tolerance when the tool axis orientation is smoothed. In the paper, the bottom tool path segments are first smoothed by Bezier spline with an optimized feedrate while respecting corner error tolerance and tangential kinematic limits. Then, the top tool axis orientation path segments are smoothed with additional feedrate synchronization constraint. At last, the tool axis orientation and tool tip positions are synchronized to avoid discontinuous displacements along the tool axis, and the maximum feedrate allowed on the bottom corner transition spline would be further limited when the bottom or top corner transition spline is revised. The effectiveness of the proposed method to smoothen the five-axis tool paths have been demonstrated with simulations and experiments.

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Abbreviations

\({Q}_{i}^{b}\) :

Tool tip position retrieved from tool path data (NC)

\({Q}_{i}^{t}\) :

Tool axis orientation vector retrieved from tool path data (NC)

\({B}_{i}^{b}\) :

Bezier spline inserted into the corner defined by points \(\left({Q}_{i-1}^{b},{Q}_{i}^{b},{Q}_{i+1}^{b}\right)\)

\({V}_{i}^{b}\) :

Maximum allowed feedrate of Bezier spline \({B}_{i}^{b}\)

\({P}_{i,m}^{b}\left(m=0,...,6\right)\) :

Control points of Bezier spline \({B}_{i}^{b}\)

\(\eta\) :

Angle \(\mathrm{\angle }{Q}_{i-1}^{b}{Q}_{i}^{b}{Q}_{i+1}^{b}\)

\({l}_{i}^{b}\) :

Transition length of Bezier \({B}_{i}^{b}\)  evaluated as the length of \(\overrightarrow{{P}_{i,0}^{b}{P}_{i,1}^{b}}\)

\(C(u)\) :

Bezier spline

\({C}^{b}\left(u\right),{C}^{t}(u)\) :

Tool tip position Bezier spline and tool axis orientation Bezier spline, respectively

\(P_i\) :

Control points of \(C(u)\) at \(i\)

\({b}_{i,n}(u)\) :

Basis functions of \(C(u)\)

u :

Spline parameter \((0\le u\le 1)\)  

\({\kappa }_{i,\mathrm{max}}^{b}\) :

Maximum curvature of the Bezier spline \({B}_{i}^{b}\)

\({e}_{trans}^{b},{e}_{ch}^{b}\) :

Corner transition error tolerance and chord error tolerance of tool tip position path, respectively

\(F,{A}_{\mathrm{max}},{J}_{\mathrm{max}}\) :

Feedrate command, tangential acceleration, and jerk limits, respectively

\({V}_{ch,i}^{b},{V}_{Acc,i}^{b},{V}_{J,i}^{b}\) :

Maximum allowed feedrate under the constraints of \({e}_{ch}^{b}\), \({A}_{\mathrm{max}}\), and \({J}_{\mathrm{max}}\), respectively

T :

Interpolation period

\({\delta }_{trans,i}^{b}\) :

\({\delta }_{trans,i}=\Vert {Q}_{i}^{b}{B}_{i}^{b}\left(0.5\right)\Vert\) is the corner transition error of Bezier spline \({B}_{i}^{b}\)

\({\delta }_{ch,i}^{b}\) :

Chord error during the interpolation of Bezier spline \({B}_{i}^{b}\)

\({\delta }_{CE,i}^{b}\) :

Corner error of corner \({Q}_{i-1}^{b}{Q}_{i}^{b}{Q}_{i+1}^{b}\)

\({e}_{CE}^{b}\) :

\({e}_{CE}^{b}={e}_{trans}^{b}+{e}_{ch}^{b}\) is the corner error tolerance of tool tip position path

\({g}_{1}\left(\kappa \right),{g}_{2}\left(\kappa \right)\) :

Functions of \({V}_{ch,i}^{b}-F\), \({V}_{ch,i}^{b}-{V}_{J,i}^{b}\), respectively

\({N}_{{g}_{2}(\kappa)}\) :

Number of real roots of function \({g}_{2}\left(\kappa \right)=0\)  

\({\kappa }_{i,\mathrm{max},k}^{b}\left(k=1,...,6\right)\) :

Curvature of candidate feedrates of \({V}_{i}^{b}\)

\({g}_{3}\left(\kappa \right)\) :

Maximum allowed feedrate of Bezier spline \({B}_{i}^{b}\) at curvature parameter \(\kappa\)

\({d}_{i,pre}^{b},{d}_{i,after}^{b}\) :

Length of linear segments \({Q}_{i-1}^{b}{Q}_{i}^{b}\), \({Q}_{i}^{b}{Q}_{i+1}^{b}\), respectively

\({e}_{orientation}\) :

Tool axis orientation error tolerance

\({u}_{j},{u}_{j+1}\) :

Parameters of Bezier spline \({B}_{i}^{b}\)

\({B}_{i}^{t}\) :

Bezier spline inserted into the corner of tool axis orientation path defined by points \(\left({Q}_{i-1}^{t},{Q}_{i}^{t},{Q}_{i+1}^{t}\right)\)

\({P}_{i,middle}^{b},{P}_{i,middle}^{t}\) :

Middle points of linear segments \({B}_{i}^{b}\left({u}_{j}\right){B}_{i}^{b}\left({u}_{j+1}\right)\) and \({B}_{i}^{t}\left({u}_{j}\right){B}_{i}^{t}\left({u}_{j+1}\right)\), respectively

\({\delta }_{CE,i}^{t}\) :

Corner error of corner \({Q}_{i-1}^{t}{Q}_{i}^{t}{Q}_{i+1}^{t}\)

\({e}_{CE}^{t}\) :

\({e}_{CE}^{t}={e}_{trans}^{t}+{e}_{ch}^{t}\) is the corner error tolerance of tool axis orientation path

\({e}_{ch,i}^{b},{e}_{ch,i}^{t}\) :

Revised chord error tolerance of Bezier splines \({B}_{i}^{b}\) and \({B}_{i}^{t}\), respectively

\(O\) :

Center of the circle of curvature at point \({B}_{i}^{b}\left(0.5\right)\)

\(\alpha\) :

Half of the central angle \(\angle {B}_{i}^{b}\left({u}_{j}\right)O{B}_{i}^{b}\left({u}_{j+1}\right)\)

\(\overset\frown{B_i^b\left(u_j\right)B_i^b\left(u_{j+1}\right)}\) :

Arc length along Bezier spline \({B}_{i}^{b}\) in a single interpolation period at point \({B}_{i}^{b}\left(0.5\right)\)  

\({C}^{b{^{\prime}}},{C}^{b{^{\prime\prime}}}\) :

The first and second derivatives of tool tip position path, respectively

\({P}_{i,m}^{t}\left(m=0,...,6\right)\) :

Control points of Bezier spline \({B}_{i}^{t}\) 

\({l}_{i}^{t}\) :

Transition length of Bezier \({B}_{i}^{t}\), which is evaluated as the length of \(\overrightarrow{{P}_{i,0}^{t}{P}_{i,1}^{t}}\)  

\({S}_{spline,i}^{t}\left({l}_{i}^{t},{u}_{1},{u}_{2}\right)\) :

Arc length along Bezier spline \({B}_{i}^{t}\)  in parameter interval \(\left[{u}_{1},{u}_{2}\right]\)

\({v}_{i}^{t}\) :

Feedrate on the tool axis orientation spline \({B}_{i}^{t}\)  in parameter interval \(\left[{u}_{1},{u}_{2}\right]\)

\(\theta\) :

Angle \(\mathrm{\angle }{Q}_{i-1}^{t}{Q}_{i}^{t}{Q}_{i+1}^{t}\)

\({e}_{ch}^{t}\) :

Chord error tolerance of tool axis orientation path

\(\begin{array}{r}dOri/ds,\,{d}^{2}Ori/d{s}^{2},\\\,{d}^{3}Ori/d{s}^{3}\end{array}\) :

Rate of change in tool axis orientation with respect to the tool tip displacement

\({u}_{s},{u}_{ss},{u}_{sss}\) :

The first, second, and third derivatives of u with respect to the tool tip displacement

\({C}^{t{^{\prime}}},\,{C}^{t{^{\prime\prime}}},\,{C}^{{t^{\prime\prime\prime}}}\) :

The first, second, and third derivatives of tool axis orientation path, respectively

\({u}_{extreme,k}\left(k=1,...,N\right)\) :

Parameters of maximum curvature points on the final constructed Bezier splines (\({B}_{i}^{b}\), \({B}_{i}^{t}\))

\({O}_{extreme,1}^{t}\) :

Center of the circle of curvature at point \({B}_{i}^{t}\left({u}_{extreme,1}\right)\)

\({\alpha }_{extreme,1}^{t}\) :

Half of the central angle \(\mathrm{\angle }{B}_{i}^{t}\left({u}_{r}\right){O}_{extreme,1}^{t}{B}_{i}^{t}\left({u}_{r+1}\right)\)

\({V}_{{u}_{extreme,k}}\left(k=1,...,N\right)\) :

The maximum allowed feedrate of Bezier spline \({B}_{i}^{b}\)  at point \({B}_{i}^{b}\left({u}_{extreme,k}\right)\)

\(\begin{array}{r}\left({d}^{3}X/d{s}^{3},\right.{d}^{3}Y/d{s}^{3},\\ \left.{d}^{3}Z/d{s}^{3}\right)\end{array}\) :

The third geometric derivatives of the tool tip \({Q}^{b}(X,Y,Z)\)  with respect to displacement

\(\begin{array}{l}\left({d}^{3}i/d{s}^{3},\right.{d}^{3}j/d{s}^{3},\\ \left.{d}^{3}k/d{s}^{3}\right)\end{array}\) :

The third geometric derivatives of the tool axis orientation \(Ori(i,j,k)\)  with respect to displacement

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Acknowledgements

Prof. Yusuf Altintas supervised and supported the research. The author also thanks Atakan Zeybek, a master student at Manufacturing Automation Laboratories (MAL) of the University of British Columbia (UBC), Canada, for supporting experimental portion of this work.

Funding

This research is sponsored by the Chinese Academy of Sciences in the form of Postdoctoral Fellowship to Sun Shujie (National Natural Science Foundation of China-NSFC, 62002308; China Scholarship Council; China Postdoctoral Science Foundation, 2018M641729; Shandong Provincial Natural Science Foundation, china, ZR2018BEE019), and NSERC Canadian Network for Research and Innovation in Machining Technology Network Phrase 2 (CANRIMT2).

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Authors and Affiliations

  1. Manufacturing Automation Laboratory, The University of British Columbia, Vancouver, BC, V6T1Z4, Canada

    Shujie Sun

  2. School of Electromechanical and Automotive Engineering, Yantai University, Yantai, 264005, China

    Shujie Sun

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  1. Shujie Sun
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Correspondence to Shujie Sun.

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Appendices

Appendix 1

The intersection points of F and \({V}_{ch,i}^{b}\) are illustrated in Fig. 16 and \({\kappa }_{i,\mathrm{max},1}^{b}\) is evaluated as the following logic:

Fig. 16
figure 16

Intersection points of \(F\) and \({V}_{ch,i}^{b}\). a No solution. b 1 solution. c 2 solutions

Full size image

  • If \({N}^{2}-4MH<0\) (Fig. 16a), \({g}_{1}\left(\kappa \right)=0\) does not have real roots, and the curvature for \({V}_{i}^{b}\) is \({\kappa }_{3}\), i.e., \({\kappa }_{i,\mathrm{max},1}^{b}=-2M/N\).

  • If \({N}^{2}-4MH=0\) (Fig. 16b), \({g}_{1}\left(\kappa \right)=0\) has only one real root, and the curvature for \({V}_{i}^{b}\) is \({\kappa }_{3}\), i.e., \({\kappa }_{i,\mathrm{max},1}^{b}=-2M/N\).

  • If \({N}^{2}-4MH>0\) (Fig. 16c), \({g}_{1}\left(\kappa \right)=0\) has two real roots, and the curvature for \({V}_{i}^{b}\) is the minimum real root of \({g}_{1}\left(\kappa \right)=0\), i.e., \({\kappa }_{i,\mathrm{max},1}^{b}=\frac{2M}{-N-\sqrt{{N}^{2}-4MH}}\).

Hence, \({\kappa }_{i,\mathrm{max},1}^{b}\) is evaluated as:

$$\kappa_{i,\max,1}^b=\left\{\begin{array}{c}-2M/N\;\;\;\;if\;\;N^2-4MH\le0\;\;\;\;\;\;\;\;\;\;\\\frac{2M}{-N-\sqrt{N^2-4MH}}\;\;\;\;if\;\;N^2-4MH>0\end{array}\right.$$
(43)

Appendix 2

The number of distinct real roots of \({g}_{2}\left(\kappa \right)\) is evaluated by the Sturm’s theorem [29]. The Sturm sequence of \({g}_{2}\left(\kappa \right)\) is the sequence of polynomials \({f}_{i}\left(x\right)\left(i=0,...,6\right)\) and evaluated as:

$${f}_{0}\left(x\right)={e}_{6}{x}^{6}+{e}_{5}{x}^{5}+{e}_{4}{x}^{4}+{e}_{3}{x}^{3}+{e}_{2}{x}^{2}+{e}_{1}{x}^{1}+{e}_{0}{x}^{0}$$
(44)

where \({e}_{6}={M}^{3}\), \({e}_{5}=3{M}^{2}N\), \({e}_{4}=3M{N}^{2}-3{M}^{2}{\left({e}_{CE}^{b}\right)}^{2}-\frac{{J}_{\mathrm{max}}^{2}{T}^{6}}{{2}^{6}}\), \({e}_{3}=N\left({N}^{2}-6M{\left({e}_{CE}^{b}\right)}^{2}\right)\), \({e}_{2}=3{\left({e}_{CE}^{b}\right)}^{2}\left(M{\left({e}_{CE}^{b}\right)}^{2}-{N}^{2}\right)\), \({e}_{1}=3N{\left({e}_{CE}^{b}\right)}^{4}\), \({e}_{0}=0\), \(x=\frac{1}{{\kappa }_{i,\mathrm{max},6}^{b}}\)

$${f}_{1}\left(x\right)={f}_{5}{x}^{5}+{f}_{4}{x}^{4}+{f}_{3}{x}^{3}+{f}_{2}{x}^{2}+{f}_{1}{x}^{1}+{f}_{0}={f}^{^{\prime}}\left(x\right)$$
(45)

where \({f}_{5}=6{e}_{6},{f}_{4}=5{e}_{5},{f}_{3}=4{e}_{4},{f}_{2}=3{e}_{3},{f}_{1}=2{e}_{2},{f}_{0}={e}_{1}\)

$${f}_{2}\left(x\right)={g}_{4}{x}^{4}+{g}_{3}{x}^{3}+{g}_{2}{x}^{2}+{g}_{1}{x}^{1}+{g}_{0}=-rem\left({f}_{0}\left(x\right),{f}_{1}\left(x\right)\right)$$
(46)

where \({g}_{4}=a{f}_{3}+b{f}_{4}-{e}_{4}\), \({g}_{3}=a{f}_{2}+b{f}_{3}-{e}_{3}\), \({g}_{2}=a{f}_{1}+b{f}_{2}-{e}_{2}\), \({g}_{1}=a{f}_{0}+b{f}_{1}-{e}_{1}\), \({g}_{0}=b{f}_{0}-{e}_{0}\), \(a=\frac{{e}_{6}}{{f}_{5}}\), \(b=\frac{{e}_{5}-a{f}_{4}}{{f}_{5}}\)

$${f}_{3}\left(x\right)={h}_{3}{x}^{3}+{h}_{2}{x}^{2}+{h}_{1}{x}^{1}+{h}_{0}=-rem\left({f}_{1}\left(x\right),{f}_{2}\left(x\right)\right)$$
(47)

where \({h}_{3}=c{g}_{2}+d{g}_{3}-{f}_{3}\), \({h}_{2}=c{g}_{1}+d{g}_{2}-{f}_{2}\), \({h}_{1}=c{g}_{0}+d{g}_{1}-{f}_{1}\), \({h}_{0}=d{g}_{0}-{f}_{0}\), \(d=\frac{{f}_{4}-c{g}_{3}}{{g}_{4}}\), \(c=\frac{{f}_{5}}{{g}_{4}}\),

$${f}_{4}\left(x\right)={i}_{2}{x}^{2}+{i}_{1}x+{i}_{0}=-rem\left({f}_{2}\left(x\right),{f}_{3}\left(x\right)\right)$$
(48)

where \({i}_{2}=e{h}_{1}+f{h}_{2}-{g}_{2}\), \({i}_{1}=e{h}_{0}+f{h}_{1}-{g}_{1}\), \({i}_{0}=f{h}_{0}-{g}_{0}\), \(f=\frac{{g}_{3}-e{h}_{2}}{{h}_{3}}\), \(e=\frac{{g}_{4}}{{h}_{3}}\)

$${f}_{5}\left(x\right)={j}_{1}x+{j}_{0}=-rem\left({f}_{3}\left(x\right),{f}_{4}\left(x\right)\right)$$
(49)

where \({j}_{1}=g{i}_{0}+h{i}_{1}-{h}_{1}\), \({j}_{0}=h{i}_{0}-{h}_{0}\), \(h=\frac{{h}_{2}-g{i}_{1}}{{i}_{2}}\), \(g=\frac{{h}_{3}}{{i}_{2}}\)

$${f}_{6}\left(x\right)={k}_{0}=-rem\left({f}_{4}\left(x\right),{f}_{5}\left(x\right)\right)$$
(50)

where \({k}_{0}=j{j}_{0}-{i}_{0}\), \(j=\frac{{i}_{2}-i{j}_{0}}{{j}_{1}}\), \(i=\frac{{i}_{2}}{{j}_{1}}\)

The number of sign variation at parameter \(\kappa\) of the Sturm sequence of \({g}_{2}\left(\kappa \right)\) is donated here as \(signal\_change\left(\kappa \right)\). Hence, the number of distinct real roots of \({g}_{2}\left(\kappa \right)\) in the curvature interval \(\left[{\kappa }_{2},{\kappa }_{1}\right]\) is:

$$\begin{array}{c}signal\_change\left({f}_{0}\left(^1/_{\kappa_1}\right),{f}_{1}\left(^1/_{\kappa_{1}}\right),{f}_{2}\left(^1/_{\kappa_{1}}\right),{f}_{3}\left(^1/_{\kappa_{1}}\right),{f}_{4}\left(^1/_{\kappa_{1}}\right),{f}_{5}\left(^1/_{\kappa_{1}}\right),{f}_{6}\left(^1/_{\kappa_{1}}\right)\right)-\\ signal\_change\left({f}_{0}\left(^1/_{\kappa_{2}}\right),{f}_{1}\left(^1/_{\kappa_{2}}\right),{f}_{2}\left(^1/_{\kappa_{2}}\right),{f}_{3}\left(^1/_{\kappa_{2}}\right),{f}_{4}\left(^1/_{\kappa_{2}}\right),{f}_{5}\left(^1/_{\kappa_{2}}\right),{f}_{6}\left(^1/_{\kappa_{2}}\right)\right)\end{array}$$
(51)

Appendix 3

The first, second, and third derivatives of \(C(u)\) (Eq. 3) are [20]:

$$\left.\begin{array}{c}{C^{^{\prime}}}(u)=n\sum\limits_{i=0}^{n-1}{b}_{i,n-1}(u)({P}_{i\text{+}1}-{P}_{i})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {C^{^{\prime\prime}}}(u)=n(n-1)\sum\limits_{i=0}^{n-2}{b}_{i,n-2}(u)({P}_{i+2}-2{P}_{i+1}+{P}_{i})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ {C^{^{\prime\prime\prime}}}(u)=n(n-1)(n-2)\sum\limits_{i=0}^{n-3}{b}_{i,n-3}(u)({P}_{i+3}-3{P}_{i+2}+3{P}_{i+1}-{P}_{i})\end{array}\right\}$$
(52)

As shown in Fig. 5, the first, second, and third derivatives of splines \({B}_{i}^{b}\) and \({B}_{i}^{t}\) at their starting points (\(u=0\)) can be evaluated as \({C}^{b^{\prime}}(0)=6l\overrightarrow{{Q}_{i-1}^{b}{Q}_{i}^{b}}/\Vert \overrightarrow{{Q}_{i-1}^{b}{Q}_{i}^{b}}\Vert\), \({C}^{b^{\prime\prime}}(0)=0\), \({C}^{b^{\prime\prime\prime}}(0)\), and \({C}^{t^{\prime}}(0)=6l\overrightarrow{{Q}_{i-1}^{t}{Q}_{i}^{t}}/\Vert \overrightarrow{{Q}_{i-1}^{t}{Q}_{i}^{t}}\Vert\), \({C}^{t^{\prime\prime}}(0)=0\), \({C}^{t^{\prime\prime\prime}}(0)=0\), respectively. Combining Eq. (38), \({d}^{2}Ori/d{s}^{2}\), \({d}^{3}Ori/d{s}^{3}\) of the transition spline (\({B}_{i}^{b}\)) at the starting point (\({P}_{i,0}^{b}\)) and \({d}^{2}Ori/d{s}^{2}\), \({d}^{3}Ori/d{s}^{3}\) of tool path linear segment \({Q}_{i-1}^{b}{P}_{i,0}^{b}\) at the end point (\({P}_{i,0}^{b}\)) are zero, which indicates \({d}^{2}Ori/d{s}^{2}\) and \({d}^{3}Ori/d{s}^{3}\) are continuous at point \({P}_{i,0}^{b}\). Thus, it is only necessary to ensure \(dOri/ds\) is continuous at connection point \({P}_{i,0}^{b}\).

The first derivative of tool axis orientation of the linear segment \({Q}_{i-1}^{b}{P}_{i,0}^{b}\) at its end point \({P}_{i,0}^{b}\) (\(u=1\)) is evaluated as:

$$\frac{dOr{i}_{i-1}(u=1)}{ds}=\frac{{C}^{t^{\prime}}(1)}{\Vert {C}^{t}(1)\Vert }\frac{1}{\Vert {C}^{b^{\prime}}(1)\Vert }$$
(53)

Since \({C}^{t^{\prime}}\left(1\right)={P}_{i,0}^{t}={Q}_{i-1}^{t}\), \({C}^{b^{\prime}}\left(1\right)={P}_{i,0}^{b}={Q}_{i-1}^{b}\), and \({C}^{t}(1)={P}_{i,0}^{t}\), Eq. (53) can be reorganized as:

$$\frac{dOr{i}_{i-1}(u=1)}{ds}=\frac{{P}_{i,0}^{t}-{Q}_{i-1}^{t}}{\Vert {P}_{i,0}^{t}\Vert }\frac{1}{\Vert {P}_{i,0}^{b}-{Q}_{i-1}^{b}\Vert }$$
(54)

For the transition Bezier curve \({B}_{i}^{b}\), the first derivative of tool axis orientation at the starting point (\(u=0\)) is evaluated as:

$$\frac{dOr{i}_{i}(u=0)}{ds}=\frac{{C}^{t^{\prime}}(0)}{\Vert {C}^{t}(0)\Vert }\frac{1}{\Vert {C}^{b^{\prime}}(0)\Vert }$$
(55)

Since \({C}^{t^{\prime}}\left(0\right)=6({P}_{i,1}^{t}-{P}_{i,0}^{t})\), \({C}^{b^{\prime}}\left(0\right)=6({P}_{i,1}^{b}-{P}_{i,0}^{b})\), and \({C}^{t}(0)={P}_{i,0}^{t}\), Eq. (55) can be reorganized as:

$$\frac{dOr{i}_{i}(u=0)}{ds}=\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}}{\Vert {P}_{i,0}^{t}\Vert }\frac{1}{\Vert {P}_{i,1}^{b}-{P}_{i,0}^{b}\Vert }$$
(56)

\(dOri/ds\) must match at the connection point \({P}_{i,0}^{b}\) (i.e.\(dOr{i}_{i-1}(u=1)/ds\text{=}dOr{i}_{i}(u=0)/ds\)), which yields:

$$\frac{{P}_{i,0}^{t}-{Q}_{i-1}^{t}}{{P}_{i,0}^{b}-{Q}_{i-1}^{b}}=\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}}{{P}_{i,1}^{b}-{P}_{i,0}^{b}}$$
(57)

Considering,

$$\left\{\begin{array}{c}\Vert {P}_{i,1}^{b}-{P}_{i,0}^{b}\Vert =\Vert {P}_{i,2}^{b}-{P}_{i,1}^{b}\Vert =\Vert {Q}_{i}^{b}-{P}_{i,2}^{b}\Vert \\ \Vert {P}_{i,1}^{t}-{P}_{i,0}^{t}\Vert =\Vert {P}_{i,2}^{t}-{P}_{i,1}^{t}\Vert =\Vert {Q}_{i}^{t}-{P}_{i,2}^{t}\Vert \end{array}\right.$$
(58)

Therefore,

$$\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}}{{P}_{i,1}^{b}-{P}_{i,0}^{b}}=\frac{{P}_{i,2}^{t}-{P}_{i,1}^{t}}{{P}_{i,2}^{b}-{P}_{i,1}^{b}}=\frac{{Q}_{i}^{t}-{P}_{i,2}^{t}}{{Q}_{i}^{b}-{P}_{i,2}^{b}}$$
(59)

Combing Eq. (57),

$$\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}}{{P}_{i,1}^{b}-{P}_{i,0}^{b}}\text{=}\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}+{P}_{i,2}^{t}-{P}_{i,1}^{t}+{Q}_{i}^{t}-{P}_{i,2}^{t}\text{+}{P}_{i,0}^{t}-{Q}_{i-1}^{t}}{{P}_{i,1}^{b}-{P}_{i,0}^{b}+{P}_{i,2}^{b}-{P}_{i,1}^{b}+{Q}_{i}^{b}-{P}_{i,2}^{b}\text{+}{P}_{i,0}^{b}-{Q}_{i-1}^{b}}$$
(60)

Considering,

$$\left\{\begin{array}{c}{P}_{i,1}^{t}-{P}_{i,0}^{t}+{P}_{i,2}^{t}-{P}_{i,1}^{t}+{Q}_{i}^{t}-{P}_{i,2}^{t}+{P}_{i,0}^{t}-{Q}_{i-1}^{t}={Q}_{i}^{t}-{Q}_{i-1}^{t}\\ {P}_{i,1}^{b}-{P}_{i,0}^{b}+{P}_{i,2}^{b}-{P}_{i,1}^{b}+{Q}_{i}^{b}-{P}_{i,2}^{b}+{P}_{i,0}^{b}-{Q}_{i-1}^{b}={Q}_{i}^{b}-{Q}_{i-1}^{b}\end{array}\right.$$
(61)

Equation (60) leads to

$$\frac{{Q}_{i}^{t}-{Q}_{i-1}^{t}}{{Q}_{i}^{b}-{Q}_{i-1}^{b}}=\frac{{P}_{i,1}^{t}-{P}_{i,0}^{t}}{{P}_{i,1}^{b}-{P}_{i,0}^{b}}$$
(62)

Similarly,

$$\frac{{Q}_{i}^{t}-{Q}_{i+1}^{t}}{{Q}_{i}^{b}-{Q}_{i+1}^{b}}=\frac{{P}_{i,6}^{t}-{P}_{i,5}^{t}}{{P}_{i,6}^{b}-{P}_{i,5}^{b}}$$
(63)

Hence, the inserted transition Bezier spline \({B}_{i}^{b}\) which has continuous orientation yields Eq. (39).

Appendix 4

Similar with Eq. (6), the transition error tolerance of spline \({B}_{i}^{t}\) is evaluated as:

$${e}_{trans,i}^{t}=15{l}_{i}^{t}\sqrt{2\left(1+\mathrm{cos}\theta \right)}/32$$
(64)

where \({l}_{i}^{t}\) is obtained from Eq. (37).

The chord error tolerance is obtained as follows:

$${e}_{ch,i}^{t}={e}_{CE}^{t}-{e}_{trans,i}^{t}$$
(65)

Considering curvature at point \({B}_{i}^{t}\left({u}_{extreme,1}\right)\),

$${\kappa }_{{u}_{extreme,1}}^{t}\text{=}\frac{\Vert {B}^{t^{\prime}}({u}_{extreme,1})\times {B}^{t^{\prime\prime}}({u}_{extreme,1})\Vert }{{\Vert {B}^{t^{\prime}}({u}_{extreme,1})\Vert }^{3}}$$
(66)

The arc length along spline \({B}_{i}^{t}\) in a single interpolation period at point \({B}_{i}^{t}\left({u}_{extreme,1}\right)\) can be evaluated by the local approximation method as (Fig. 6):

$$\parallel\overset\frown{B_i^t\left(u_r\right)B_i^t\left(u_{r+1}\right)}\parallel=\frac2{\kappa_{u_{extreme,1}}^t}arcos\frac{1/\kappa_{u_{extreme,1}}^t-e_{ch,i}^t}{1/\kappa_{u_{extreme,1}}^t}$$
(67)

Therefore, the parameter \({u}_{r+1}\) is evaluated as:

$${u}_{r+1}=u\left({s}_{{u}_{extreme,1}}^{t}\right)+{\left.\frac{du}{ds}\right|}_{S={S}_{{u}_{extreme,1}}^{t}}\Delta {s}_{j}+{\left.\frac{1}{2}\frac{{d}^{2}u}{d{s}^{2}}\right|}_{S={S}_{{u}_{extreme,1}}^{t}}\Delta {s}_{j}^{2}$$
(68)

where

$$u\left(s_{u_{extreme,1}}^t\right)=u_{extreme,1},\Delta s_j=\parallel\overset\frown{B_i^t\left(u_r\right)B_i^t\left(u_{r+1}\right)}\parallel/2,$$
$$\frac{du}{ds}=\frac{du}{\sqrt{{d}_{x}^{2}+{d}_{y}^{2}+{d}_{z}^{2}}}=\frac{1}{\sqrt{x{(u)}^{^{\prime}{2}}+y{(u)}^{^{\prime}{2}}+z{(u)}^{^{\prime}{2}}}}=\frac{1}{\Vert {C}^{t^{\prime}}(u)\Vert }, \frac{{d}^{2}u}{d{s}^{2}}=-\frac{\langle {C}^{t^{\prime}}\left(u\right),{C}^{t^{\prime\prime}}\left(u\right)\rangle }{{\Vert {C}^{t^{\prime}}\left(u\right)\Vert }^{4}},$$
$${x}^{^{\prime}}\left(u\right)=\sum_{i=0}^{6}{b}_{i,n}^{^{\prime}}(u){P}_{i,x}^{t}, {y}^{^{\prime}}\left(u\right)=\sum_{i=0}^{6}{b}_{i,n}^{^{\prime}}(u){P}_{i,y}^{t}, {z}^{^{\prime}}\left(u\right)=\sum_{i=0}^{6}{b}_{i,n}^{^{\prime}}(u){P}_{i,z}^{t}.$$

Since point \({u}_{r}\) and \({u}_{r+1}\) are symmetric about point \({u}_{extreme,1}\), the parameter interval of arc \(\Arrowvert\overset\frown{B_i^t\left(u_r\right)B_i^t\left(u_{r+1}\right)}\Arrowvert\) is \(\left[2{u}_{extreme,1}-{u}_{r+1},{u}_{r+1}\right]\).

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Sun, S. A G3 continuous five-axis tool path corner smoothing method with improved machining efficiency and accurately controlled deviation of tool axis orientation. Int J Adv Manuf Technol 119, 7003–7024 (2022). https://doi.org/10.1007/s00170-021-08227-4

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