An analytical G3 continuous corner smoothing method with adaptive constraints adjustments for five-axis machine tool

Abstract

In this paper, an analytical local corner smoothing method is proposed with two specially designed quartic curves to generate smooth motion trajectories of five-axis linear segments. One symmetrical quartic B-spline curve is used to smooth the tool tip position in workpiece coordinate system (WCS); meanwhile, another asymmetrical quartic B-spline curve is used to smooth the tool orientation in machine coordinate system (MCS). With the proposed method, not only the transition error of tool tip position and the smoothing error of the tool orientation are mathematically constrained in WCS, but the G3 continuity of the tool tip position and tool orientation are guaranteed along the entire toolpath. Besides, the maximum possible machining error of tool tip position caused by transition error and chord error is also controlled analytically by transition error adjustment. What’s more, in feedrate scheduling, the feedrate of tool tip is moderately adjusted by smooth feedrate profile, which can limit the maximum feedrate of tool orientation and achieve continuous axial feedrate and acceleration. Finally, the effectiveness and feasibility of the proposed method have been validated via simulations.

Appendices

The analytical solution of the curvature extreme

According the predefined knot vector, the basis function can be calculated as,

$$ \left\{\begin{array}{l}{N}_{0,4}={\left(2u-1\right)}^4\\ {}{N}_{1,4}=-8u\cdotp {\left(2u-1\right)}^3\\ {}\begin{array}{l}{N}_{2,4}=4{u}^2\cdotp \left(17{u}^2-20u+6\right)\\ {}{N}_{3,4}=-8{u}^3\cdotp \left(3u-2\right)\\ {}{N}_{4,4}=4{u}^4\\ {}\begin{array}{c}{N}_{5,4}=0\\ {}{N}_{6,4}=0\end{array}\end{array}\end{array}\right.\kern0.5em ,u\in \left[\mathrm{0.5,1}\right)\ \mathrm{and}\kern0.5em \left\{\begin{array}{l}\begin{array}{l}\begin{array}{l}{N}_{0,4}=0\\ {}{N}_{1,4}=0\\ {}{N}_{2,4}=4{\left(u-1\right)}^4\end{array}\\ {}{N}_{3,4}=-8\left(3u-1\right)\cdotp {\left(u-1\right)}^3\\ {}{N}_{4,4}=4{\left(u-1\right)}^2\cdotp \left(17{u}^2-14u+3\right)\end{array}\\ {}{N}_{5,4}=-8{\left(2u-1\right)}^3\cdotp \left(u-1\right)\\ {}{N}_{6,4}={\left(2u-1\right)}^4\end{array}\right.\kern0.5em ,u\in \left[\mathrm{0.5,1}\right] $$

(47)

And a blending B-spline curve illustrated in Fig. 21 is constructed, where a = b = l_p. The control points can be written as,

$$ {\displaystyle \begin{array}{l}{\boldsymbol{P}}_0=\left(2{l}_p,0\right);\\ {}{\boldsymbol{P}}_1=\left(1.5{l}_p,0\right);\\ {}{\boldsymbol{P}}_2=\left({l}_p,0\right);\\ {}{\boldsymbol{P}}_3=\left(0,0\right);\\ {}{\boldsymbol{P}}_4=\left({l}_p\cdotp \cos \left(\theta \right),{l}_p\cdotp \sin \left(\theta \right)\right);\\ {}{\boldsymbol{P}}_5=\left(1.5{l}_p\cdotp \cos \left(\theta \right),1.5{l}_p.\sin \left(\theta \right)\right);\\ {}{\boldsymbol{P}}_6=\left(2{l}_p\cdotp \cos \left(\theta \right),2{l}_p\cdotp \sin \left(\theta \right)\right)\end{array}} $$

(48)

Fig. 21

The blending curve of a special corner

Thus, the expression of the curvature can be calculated as:

$$ k=\left\{\begin{array}{l}\frac{3{u}^2{\cos}^2\left(0.5\theta \right)\sin \left(0.5\theta \right)}{\varepsilon \cdotp \Big[{\left(1-8{u}^3\cdotp {\cos}^2\left(o.5\theta \right)\right)}^2+64{u}^6\cdotp {\cos}^2\left(0.5\theta \right)},u\in \left[\mathrm{0,0.5}\right)\\ {}\cdotp \sin \left(0.5\theta \right)\Big]{}^{\frac{3}{2}}\\ {}\frac{3{\left(u-1\right)}^2\cdotp {\cos}^2\left(0.5\theta \right)\cdotp \sin \left(0.5\theta \right)}{\begin{array}{c}\left(\varepsilon \right[4\left({\cos}^2\left(0.5\theta \right)\cdotp {\sin}^2\left(0.5\theta \right)\right)+\Big(12u-3\cos \left(0.5\theta \right)\\ {}-12{u}^2\cdotp \cos \left(0.5\theta \right)+4{u}^3\cdotp \cos \left(0.5\theta \right)+12u\cdotp \cos \left(0.5\theta \right)\\ {}-12{u}^2+4{u}^3-4\left){}^2\right]{}^{\frac{3}{2}}\Big)\end{array}}\end{array}\right. $$

(49)

Thus, the curvature at u = 0.5 can be calculated as,

$$ \kappa =\frac{3}{4\varepsilon \cdotp {\tan}^2\left(0.5\theta \right)} $$

(50)

Derivation of the smoothing error control

For convenience, the control points of the blending curve are constructed as Fig. 21 shows, and they can be written as,

$$ {\displaystyle \begin{array}{l}{\boldsymbol{P}}_0=\left(2a,0\right);\\ {}{\boldsymbol{P}}_1=\left(1.5a,0\right);\\ {}{\boldsymbol{P}}_2=\left(a,0\right);\\ {}{\boldsymbol{P}}_3=\left(0,0\right);\\ {}{\boldsymbol{P}}_4=\left(b\cdotp \cos \left(\theta \right),b\cdotp \sin \left(\theta \right)\right);\\ {}{\boldsymbol{P}}_5=\left(1.5b\cdotp \cos \left(\theta \right),1.5b\cdotp \sin \left(\theta \right)\right);\\ {}{\boldsymbol{P}}_6=\left(2b\cdotp \cos \left(\theta \right),2b\right)\cdotp \sin \left(\theta \right)\Big)\end{array}} $$

(51)

Supposing the current blending curve is C(A), the point P on the curve can be obtained as:

$$ \boldsymbol{P}=\left\{\begin{array}{l}\left[2\left(2{u}^4-2u+1\right)\cdotp \cos \left(\theta \right)\cdotp b,4{u}^4\cdotp b\right]=\left[2{f}_1\cdotp a+4{f}_2\cdotp b,4{f}_3\cdotp b\right],u\in \left[\mathrm{0,0.5}\right)\\ {}\left[4{\left(u-1\right)}^4\cdotp a+2\left(2{u}^4-8{u}^3+12{u}^2-6{u}^2-6u+1\right)\cdotp \cos \left(\theta \right)\cdotp b,2\left(2{u}^4-8{u}^3+12{u}^2-6u+1\right)\cdotp \sin \left(\theta \right)\cdotp b\right]=\left[4{f}_4\cdotp a+2{f}_5\cdotp b,2{f}_6\cdotp b\right],u\in \left[\mathrm{0.5,1}\right]\end{array}\right. $$

(52)

When $0\le \theta <\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, it is easy to prove that

$$ \left\{\begin{array}{l}{f}_1>0,{f}_2>0,{f}_3>0,u\in \left[0,0.5\right)\\ {}{f}_4>0,{f}_5>0,{f}_6>0,u\in \left[0.5,1\right]\end{array}\right. $$

(53)

Thus, when the length a or b becomes larger, the point P_b on the new blending curve C(B) and the point P on the curve C(A) satisfy the following equation:

$$ \left\{\begin{array}{l}{\boldsymbol{P}}_{b,x}-{\boldsymbol{P}}_x\ge 0\\ {}{\boldsymbol{P}}_{b,y}-{\boldsymbol{P}}_y\ge 0\end{array}\right. $$

(54)

which means that the points on curve C(B) are farther from the origin point O(P₃). Then, the theorem for $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\le \theta <\pi$ can be proved in the same way. And according to the affine invariance of the B-spline curve [29], this conclusion is still true for space curves.

The information of tool tip point and tool orientation in simulation 2

Number	Px (mm)	Py (mm)	Pz (mm)	O i	O j	O k	Number	Px (mm)	Py (mm)	Pz (mm)	O i	O j	O k
1	42.504	17.496	− 39.827	0.8464	0.1691	0.5050	22	32.939	− 16.351	− 15.272	0.9943	− 0.0531	0.0929
2	41.462	14.674	− 38.922	0.8637	0.1569	0.4789	23	33.241	− 15.695	− 15.329	0.9939	− 0.0543	0.0964
3	40.51	11.733	− 37.717	0.8810	0.1433	0.4509	24	33.776	− 14.461	− 15.92	0.9927	− 0.0555	0.1072
4	39.815	9.528	− 36.57	0.8936	0.1322	0.4288	25	34.352	− 12.981	− 16.684	0.9908	− 0.0546	0.1241
5	39.185	7.422	− 35.274	0.9055	0.1209	0.4068	26	34.911	− 11.347	− 17.555	0.9885	− 0.0497	0.1428
6	38.549	4.941	− 33.487	0.9190	0.1065	0.3795	27	35.439	− 9.551	− 18.533	0.9858	− 0.0377	0.1636
7	38.087	2.72	− 31.807	0.9304	0.0931	0.3545	28	35.951	− 7.455	− 19.718	0.9814	− 0.0174	0.191
8	37.687	0.599	− 30.126	0.9406	0.0797	0.33	29	36.386	− 5.096	− 21.086	0.9751	0.0056	0.2218
9	37.265	− 1.934	− 28.302	0.9513	0.0639	0.3016	30	36.719	− 2.399	− 22.755	0.966328	0.0291	0.2557
10	36.789	− 4.544	− 26.617	0.96085	0.0477	0.2729	31	36.88	0.279	− 24.521	0.9558	0.0522	0.2894
11	36.194	− 7.311	− 24.941	0.9696	0.0303	0.2426	32	36.928	2.853	− 26.412	0.9436	0.0746	0.3226
12	35.461	− 10.096	− 23.334	0.9772	0.0126	0.212	33	36.962	5.021	− 28.201	0.9315	0.0937	0.3514
13	34.520	− 12.880	− 21.803	0.9834	− 0.0055	0.181	34	37.007	7.168	− 30.093	0.9179	0.1125	0.3804
14	33.378	− 15.589	− 20.374	0.9884	− 0.0235	0.1502	35	37.104	9.052	− 31.799	0.9048	0.1287	0.4059
15	32.706	− 16.977	− 19.658	0.9908	− 0.0321	0.1316	36	37.293	10.942	− 33.47	0.8908	0.1442	0.4309
16	32.250	− 17.835	− 18.937	0.9923	− 0.0379	0.1173	37	37.599	12.838	− 35.038	0.8762	0.1588	0.455
17	32.018	− 18.233	− 17.905	0.9935	− 0.0428	0.1053	38	38.119	14.888	− 36.573	0.8598	0.1735	0.4802
18	32.009	− 18.236	− 17.217	0.994	− 0.0454	0.0993	39	38.755	16.865	− 37.833	0.8438	0.1865	0.5032
19	32.1	− 18.054	− 16.547	0.9944	− 0.0477	0.0945	40	40.035	20.323	− 39.464	0.8155	0.2066	0.5407
20	32.328	− 17.608	− 15.865	0.9946	− 0.05	0.0913	41	41.099	22.74	− 40.194	0.7951	0.2192	0.5656
21	32.645	− 16.967	− 15.421	0.9945	− 0.0518	0.0911