1 Introduction

Multi-axis machine tools with clear advantages of higher material removal rate, lower cutting time, fewer set-ups [1, 2], better versatility, and higher machining efficiency [3] are extensively adaptable for the machining of sculptured surfaces in aerospace, military, ships, cars, and other industries [4,5,6]. Nevertheless, multi-axis machine tools cannot offer the same machining accuracy as their three-axis machine tools due to the additional rotary axes. Since the machining accuracy of machine tools directly reflects the accuracy of the machined workpiece, unknown precision of machine tools will lead to uncertain machining accuracy and surface quality of machined workpiece [7,8,9,10,11]. Therefore, how to efficiently evaluate the performance of multi-axis machine tools is of critical importance in the manufacturing community.

Considerable research work has been devoted to detect the machining accuracy of multi-axis machine tools during the past decades worldwide [12,13,14,15]. The standard ISO 10791-7 [16], whose standard test piece consists of a circle, a square, a diamond shape contour, and a 3° sided quadrilateral, aims to supply information as wide and comprehensive as possible on tests and checks and describes a machining test to assess the machining performance of machine tools. Although the performance of three-axis machine tools can be exhibited by measuring verticality, parallelism, and circularity of the standard test piece, it is not suitable for evaluating the machining performance of multi-axis machine tools. The cone frustum machining test, which was created by The National Aerospace Standard (NAS) 979 in 1969 [17], is the best known and extensively accepted in the machine tool industry as a final performance test for five-axis machine tools. Moreover, according to reference [18], the Numerical Control Gesellschaft (NCG) test piece has been developed to characterize complex curved surfaces feature. The above test pieces are utilized to exhibit the performance of machine tools. However, since the characteristics of complex curved surfaces, such as the conversion of open and close angle, variable curvature, and thin-wall, are different from the characteristics of NAS979 frustum test piece, poor machining accuracy and unsmooth contour are frequently obtained while processing parts with sculptured surfaces, despite multi-axis machine tools having been qualified by NAS979 frustum test piece. In addition, although the NCG test piece integrates numerous characteristics associated with sculptured surfaces, it cannot be popularly applied to evaluate the machining performance of multi-axis machine tools due to its complex model and time-consuming processing.

In order to address the above problems and comprehensively exhibit the machining accuracy of multi-axis machine tools, it is of great theoretical and practical significant to develop a new test piece. Hence, a novel “S” shaped detection test piece, called S-shaped test piece, was developed by Chengdu Aircraft Industry Group [19]. S-shaped test piece has already been accepted by ISO/TC39/SC2 at the 79th ISO meeting as an additional machining test in ISO 10791-7 for testing the machining accuracy of multi-axis machine tools [20]. According to reference [19], since S-shaped test piece integrates various features of sculpture surface, such as twist angle, the processing area of open angle and closed angle conversion, and variable curvature, it is a faithful representation of sculpture surface, such as aerospace structural components, turbine blades, and the key component for the diesel engine of large ships. These features have distinct advantages in detecting the machining accuracy of multi-axis machine tools than other test pieces. Therefore, research on the ability of the S-shaped test piece for evaluating the machining performance of multi-axis machine tools is of crucial importance.

According to references [21, 22], a model of the mechanical system and servo system has been set up to find out the key dynamic machine factors and develop an algorithm to track the cause of the processed error based on S-shaped test piece. Chen et al. [23] established a model of error parameter sensitivity influencing the accuracy of the curved surface and impact analysis based on the actual machining of the S-shaped sample of complex CNC machine tools, and then the five key parameters that have a great influence on machining errors are determined. Su and Wang [24] presented a new standard using an S-shaped test piece for evaluating the machining performance of five-axis machine tools. In order to assess the kinematic errors of five-axis machine tools, Ibaraki et al. [25] formulated a relationship between geometric errors of S-shaped test piece and the machine’s kinematic errors and carried out a machining test. Jiang et al. [26] found out the cause of surface morphology abnormality at machined “S” test piece by analyzing the geometric and machining features of “S” test piece.

As can be observed in the above-mentioned studies, considerable research work has been devoted to analyze the influence of machine errors for machine tools on the machining accuracy of the S-shaped test piece. The theoretical errors existed in the design stage of the S-shaped test piece and are the natural property of S-shaped test piece. Due to the existence of twist angle, the direction vector of non-developed rule surface along the generatrix varies unevenly during the process of modeling of S-shaped test piece. And it can only ensure that the normal vector of non-developed rule surface coincides with the normal vector of tool surface at one point in the actual flank milling process of S-shaped test piece. Thus, it is obvious that the theoretical error exists while planning tool path. In addition, the S-shaped test piece formed by a thickening of the non-developed rule surface cannot be developed by the envelope of a cylinder with a radius greater than zero. Therefore, the theoretical error cannot be averted so long as the radius of the tool is not equal to zero.

Due to the presence of theoretical error, this interference that theoretical errors have some influence on the results of the analysis introduces mistakes, when manufacturers of machine tools use the S-shaped test piece to assess the machining performance of machine tools. The performance of multi-axis machine tools cannot be comprehensively and accurately evaluated. Therefore, it is of great theoretical and practical significance to reduce the theoretical error. According to references [27, 28], the single-point offset (SPO) position method was proposed for the flank milling of the developable ruled surfaces to reduce machining error. Subsequently, in contrast to the SPO algorithm, Redonnet et al. [29, 30] introduced the three-point tangential (TPT) method to reduce machining error, and tool positions were obtained by using the CAD/CAM software. However, the machining error cannot be significantly reduced. For this reason, Bedi et al. [31] proposed the double-point tangential (DPT) method for the surface milled by using a cylindrical cutter when flank milling. According to references [32,33,34], the proposed NC path–generating algorithms are utilized for the reduction of geometrical deviations between the designed surface and the actual machine surface. Although these methods can produce a better solution in minimizing the deviations, the theoretical errors still exist. Guan et al. [35] used the single-point offset (SPO) position method to calculate the theoretical error of the S-shaped test piece and proposed pre-compensation (PRC) and post-compensation (POC) methods to decrease the influences of a theoretical error on the acceptance of multi-axis machine tools. Guan et al. [36] proposed the new three-point tangential (NTPT) algorithm for flank milling of S-shaped test piece. Experiments indicate that the proposed method has extensive adaptability for different surfaces. Nevertheless, those methods are so complex that they cannot be properly applied in the CAD/CAM software for reducing the theoretical error. Therefore, it is important to propose a feasible and effective method to reduce the theoretical error.

In view of the limitations stated, first, the uniform double cubic B-spline surface model is applied to the surface representation, and a three-dimensional model of the S-shaped test piece is established based on this model. Then, in order to analyze the influence of twist angle on the theoretical error of S-shaped test piece, the distribution of twist angle is obtained. Finally, an optimized tool path offset method, the optimized single-point offset (OSPO) method, was proposed based on the conventional single-point offset method. The optimized tool path can be subsequently generated by using CAD/CAM software on the basis of the OSPO method. With the proposed method, the theoretical error of S-shaped test piece can thus be significantly reduced. Therefore, the performance of multi-axis machine tools can be comprehensively, accurately, and effectively exhibited by S-shaped test piece.

The rest of this paper is arranged as follows. A three-dimensional model of the S-shaped test piece is established based on a uniform double cubic B-spline surface in Section 2. The influence of twist angle on the theoretical error is analyzed in Section 3. The optimized single-point offset (OSPO) method is presented and compared with traditional single-point offset method in Section 4. In Section 5, an experimental study is carried out to demonstrate the effectiveness of the proposed method, and finally some conclusions are drawn in Section 6.

2 Modeling of S-shaped test piece

As illustrated in Fig. 1, the S-shaped test piece consists of an S-shaped equal thickness edge strip and a rectangular base. Thereinto, four mounting holes with a diameter of 30 mm and two datum holes with a diameter of 16 mm are utilized to install and locate for the machining and accuracy test of S-shaped test piece respectively. The dimension and material parameters of the S-shaped test piece can be seen in Table 1. According to the requirement of United States Patent, the allowable value of machining error of the S-shaped test piece is restricted within ± 0.05 mm [19].

Fig. 1
figure 1

Engineering drawings of S-shaped test piece

Full size image
Table 1 The dimension and material parameters of the S-shaped test piece
Full size table

2.1 Directrix modeling of S-shaped test piece

At present, the B-spline surface method, Bezier surface method, Ball surface method, and Coons surface method are frequently used in CAD/CAM software. Because of geometric invariance, continuity, and symmetry, the B-spline surface method is widely applied in the field of machine design. Therefore, the uniform double cubic B-spline curve is applied to the characterization of directrix for S-shaped test piece, which is expressed as follows:

$$ Q(u)=\sum \limits_{i=0}^n{d}_i{N}_{i,k}(u) $$
(1)

where k represents the degree of B-spline curve, u represents the variable of B-spline curve (0 ≤ u ≤ 1), di represents the control points which affect the shape of the curve (i = 0, 1, 2, ..., n), and Ni, k(u) represents the basis functions with k power of u, which are the core of B-spline curve.

In addition, the Cox-de Boor method proposed in [37] is employed to establish the basis functions of the B-spline curve, so the Cox-de Boor recursion formulas of basis functions are obtained as follows:

$$ \Big\{{\displaystyle \begin{array}{l}{N}_{i,0}=\Big\{\begin{array}{l}1\kern0.75em u\in \left[{u}_i,{u}_{i+1}\right]\\ {}0\kern0.5em \mathrm{others}\end{array}\\ {}{N}_{i,k}(u)=\frac{u-{u}_i}{u_{i+k}-{u}_i}{N}_{i,k-1}(u)+\frac{u_{i+k+1}-u}{u_{i+k+1}-{u}_{i+1}}{N}_{i+1,k-1}(u)\kern0.75em k>0\\ {}\mathrm{In}\ \mathrm{particular}:\frac{0}{0}=0\end{array}} $$
(2)

Since the B-spline curve data points are subject to equidistant distribution, the basis functions are expressed as follows:

$$ \Big\{{\displaystyle \begin{array}{l}{N}_{0,3}(u)=\frac{1}{6}{\left(1-u\right)}^3\\ {}{N}_{1,3}(u)=\frac{1}{6}\left(3{u}^3-6{u}^2+4\right)\\ {}{N}_{2,3}(u)=\frac{1}{6}\left(-3{u}^3+3{u}^2+3u+1\right)\\ {}{N}_{3,3}(u)=\frac{1}{6}{u}^3\end{array}} $$
(3)

On the basis of above-mentioned equations, the uniform double cubic B-spline curve with the form of the matrix is obtained, as illustrated below:

$$ {Q}_i(u)=\frac{1}{6}\left[{u}^3\kern0.5em {u}^2\kern0.5em u\kern0.5em 1\right]\left[\begin{array}{cccc}-1& 3& -3& 1\\ {}3& -6& 3& 0\\ {}-3& 0& 3& 0\\ {}1& 4& 1& 0\end{array}\right]\left[\begin{array}{c}{d}_i\\ {}{d}_{i+1}\\ {}{d}_{i+2}\\ {}{d}_{i+3}\end{array}\right]\kern1em \left(i=1,2,\dots, n-1\right) $$
(4)

Accordingly, the first point of each segment curve can be obtained as follows:

$$ {Q}_i={Q}_i(0)=\frac{1}{6}\left({d}_i+4{d}_{i+1}+{d}_{i+2}\right)\kern0.75em i=1,2,\dots, n-1 $$
(5)

The last point of the last curve can be obtained as follows:

$$ {Q}_n={d}_{n+2} $$
(6)

The basis functions on the first two segments and the last two segments are different. The other segments have the same one. They are shown in references [21, 26]. So the following formula can be obtained:

$$ \Big\{{\displaystyle \begin{array}{l}{Q}_1={d}_1\\ {}{Q}_2=\frac{1}{4}{d}_2+\frac{7}{12}{d}_3+\frac{1}{6}{d}_4\\ {}{Q}_i=\frac{1}{6}{d}_i+\frac{2}{3}{d}_{i+1}+\frac{1}{6}{d}_{i+2}\kern0.75em \left(i=3,4,\dots, n-2\right)\\ {}{Q}_{n-1}=\frac{1}{6}{d}_{n-1}+\frac{7}{12}{d}_n+\frac{1}{4}{d}_{n+1}\\ {}{Q}_n={d}_{n+2}\end{array}} $$
(7)

Therefore, based on the above equations, control points of the S-shaped directrix are calculated, as shown in Fig. 2. Then, the S-shaped directrix can be obtained by means of substituting control points into Eq. (4). As illustrated in Fig. 3.

Fig. 2
figure 2

Control points on the directrix

Full size image
Fig. 3
figure 3

The S-shaped directrix

Full size image

2.2 S-shaped surface modeling

S-shaped surface belongs to a kind of non-developed rule surface, which was formed by generatrix moves along the upper and lower directrix as illustrated in Fig.4.

Fig. 4
figure 4

Modeling method of the S-shaped ruled surface

Full size image

So the S-shaped ruled surface is expressed as follows:

$$ \mathbf{L}\left(e,f\right)=\left(1-f\right){\mathbf{a}}_i(e)+f{\mathbf{b}}_i(e) $$
(8)

where e represents the direction variable along directrix, f represents the direction variable along generatrix, ai(e) represents the lower directrix vector, and bi(e) represents the upper directrix vector.

Meanwhile, the generatrix of the ruled surface and the mesh of ruled surface are displayed in Figs. 5 and 6, respectively.

Fig. 5
figure 5

Generatrix of ruled surface

Full size image
Fig. 6
figure 6

Mesh of ruled surface

Full size image

Then, the S-shaped test piece model can be established by means of the isometric thickening modeling method, which is illustrated in Fig. 7.

Fig. 7
figure 7

The S-shaped test piece

Full size image

3 Theoretical error analysis of S-shaped test piece

The theoretical error is caused by the existence of a twist angle and the fact that the radius of the tool is not equal to zero. As shown in Figs. 10 and 14, due to the existence of twist angle, i.e., φ, φ1, and φ2, the direction vector of non-developed rule surface along the generatrix varies unevenly during the process of modeling of S-shaped test piece, i.e., n0 ≠ n1 ≠ n2. And it can only ensure that the normal vector of non-developed rule surface coincides with the normal vector of tool surface at one point in the actual flank milling process of S-shaped test piece. Thus, it is obvious that the theoretical error cannot be averted due to the presence of twist angle in the actual flank milling processes of S-shaped test piece. Therefore, it is a prerequisite to analyze the influence of twist angle on theoretical error for improving the machining accuracy of the S-shaped test piece.

The twist angle is generated by the projection of the unit normal vector of any two directrices in the XY plane. Supposing that any two directrices of non-developed rule surface are expressed by Q1(e) = (x1(e), y1(e)) and Q2(e) = (x2(e), y2(e)), respectively. So the unit normal vectors of Q1(e) and Q2(e) are obtained, respectively, as illustrated below:

$$ {\mathbf{n}}_1=\left(\frac{y_1^{\prime }(e)}{\sqrt{\left({\left({x}_1^{\prime }(e)\right)}^2+{\left({y}_1^{\prime }(e)\right)}^2\right)}},-\frac{x_1^{\prime }(e)}{\sqrt{\left({\left({x}_1^{\prime }(e)\right)}^2+{\left({y}_1^{\prime }(e)\right)}^2\right)}}\right) $$
(9)
$$ {\mathbf{n}}_2=\left(\frac{y_2^{\prime }(e)}{\sqrt{\left({\left({x}_2^{\prime }(e)\right)}^2+{\left({y}_2^{\prime }(e)\right)}^2\right)}},-\frac{x_2^{\prime }(e)}{\sqrt{\left({\left({x}_2^{\prime }(e)\right)}^2+{\left({y}_2^{\prime }(e)\right)}^2\right)}}\right) $$
(10)

The twist angle can be obtained by considering Eqs. (9) and (10), as illustrated below:

$$ \varphi =\operatorname{arccos}\left(\frac{x_1^{\prime }(e){x}_2^{\prime }(e)+{y}_1^{\prime }(e){y}_2^{\prime }(e)}{\sqrt{{\left({x}_1^{\prime }(e)\right)}^2+{\left({y}_1^{\prime }(e)\right)}^2}+\sqrt{{\left({x}_2^{\prime }(e)\right)}^2+{\left({y}_2^{\prime }(e)\right)}^2}}\right) $$
(11)

On the basis of Eq. (11), the twist angles are obtained, as illustrated in Fig. 8. It can be seen that the twist angle rises up and down with several peaks on the whole non-developed rule surface. The maximum twist angle is approximately 6.65° at Z = 35 mm. Fig. 9 represents the regions with larger twist angle on S-shaped ruled surface.

Fig. 8
figure 8

The twist angle of S-shaped ruled surface obtained using the SPO method

Full size image
Fig. 9
figure 9

The regions with a large twist on the S-shaped ruled surface

Full size image

Flank milling principle based on the SPO method for calculating the theoretical error of the convex surface is represented in Fig. 10, where c0 denotes the directrix which is located at tool offset point, c1 denotes the lower directrix, n0, n1 denotes the unit normal vector of c0 and c1 at the cross point of two projection directrices respectively, Of denotes tool offset point, Ot denotes tool axle center, and φ denotes the twist angle, the shadow area denotes the theoretical error.

Fig. 10
figure 10

Flank milling principle based on the SPO method

Full size image

Thus, the theoretical error of convex surface at the tool offset point i can be expressed as follows:

$$ {E}_{hi}={r}_t+{r}_{si}-\sqrt{{r_{si}}^2+{r_t}^2+2{r}_{si}{r}_t\cos {\varphi}_i} $$
(12)

Similarly, the theoretical error of concave surface at the tool offset point i can be expressed as follows:

$$ {E}_{li}={r}_t-{{}^lr}_{si}+\sqrt{{{{}^lr}_{si}}^2+{r_t}^2-2{{}^lr}_{si}{r}_t\cos {\varphi}_i} $$
(13)

where rt denotes tool radius, rsi denotes curvature radius of the convex surface at tool offset point i, lrsi denotes curvature radius of the concave surface at tool offset point i, and φi denotes the twist angle at tool offset point i.

Then, the concavity and convexity of the surface can be calculated, as illustrated in Fig. 11 and Eq. 14.

$$ \Big\{{\displaystyle \begin{array}{l}\boldsymbol{\upeta} \times \boldsymbol{\uptau} >0\Rightarrow \mathrm{convex}\ \mathrm{surface}\ \mathrm{point}\\ {}\begin{array}{l}\boldsymbol{\upeta} \times \boldsymbol{\uptau} <0\Rightarrow \mathrm{concave}\ \mathrm{surface}\ \mathrm{point}\\ {}\boldsymbol{\upeta} \times \boldsymbol{\uptau} =0\Rightarrow \mathrm{plane}\ \mathrm{point}\end{array}\end{array}} $$
(14)

where τ is the tangent vector and η is the direction vector between two points. Based on the right-hand rule of directional vector cross multiplication, if the tangent vector τ is at the position of counter-clockwise direction of direction vector η, the tool offset point is the convex surface point; otherwise, it is the concave surface point.

Fig. 11
figure 11

The schematic diagram of the concavity and convexity

Full size image

Thus, the concavity and convexity of the S-shaped test piece at tool offset point are obtained, as illustrated in Fig. 12.

Fig. 12
figure 12

Concavity and convexity of S-shaped test piece

Full size image

Suppose that the tool radius is 10 mm, the single-point offset method is utilized to obtain the tool path. Then, according to Eqs. (12) and (13), the theoretical errors at different heights of S-shaped test piece are calculated, as illustrated in Fig. 13. It can be seen that the maximum theoretical errors are approximately 0.075 mm at z = 35 mm, which have exceeded the allowable value, i.e., E = 0.075mm > 0.05mm. So it is obvious that the theoretical error has significant effects when the S-shaped test piece is considered to be nearly unqualified or qualified for use. Therefore, how to develop an optimized single-point offset method to reduce theoretical errors is of paramount importance.

Fig. 13
figure 13

The theoretical errors obtained using the SPO method

Full size image

4 The optimized single-point offset (OSPO) method

In view of above-mentioned analysis, resolution strategies require a decrease in the theoretical error, and therefore, this paper proposes a method to solve this problem. In this work, the core idea of the optimized single-point offset (OSPO) method is that tool offset at one point of generatrix makes two twist angles be equal, i.e., φ1 = φ2, as illustrated in Fig. 14, where c0 denotes the directrix which is located at tool offset point, c1 denotes the lower directrix, and c2 denotes the upper directrix.

Fig. 14
figure 14

Flank milling principle based on the OSPO method

Full size image

Thus, the theoretical error of convex surface at the tool offset point i is expressed as follows:

$$ {E}_{h1i}={r}_t+{r}_{s1i}-\sqrt{{r_{s1i}}^2+{r_t}^2+2{r}_{s1i}{r}_t\cos {\varphi}_{1i}} $$
(15)
$$ {E}_{h2i}={r}_t+{r}_{s2i}-\sqrt{{r_{s2i}}^2+{r_t}^2+2{r}_{s2i}{r}_t\cos {\varphi}_{2i}} $$
(16)
$$ {E}_{hi}=\max \left({E}_{h1i},{E}_{h2i}\right) $$
(17)

Similarly, the theoretical error of concave surface at the tool offset point i is also expressed as follows:

$$ {E}_{l1i}={r}_t-{{}^lr}_{s1i}+\sqrt{{{{}^lr}_{s1i}}^2+{r_t}^2-2{{}^lr}_{s1i}{r}_t\cos {\varphi}_{1i}} $$
(18)
$$ {E}_{l2i}={r}_t-{{}^lr}_{s2i}+\sqrt{{{{}^lr}_{s2i}}^2+{r_t}^2-2{{}^lr}_{s2i}{r}_t\cos {\varphi}_{2i}} $$
(19)
$$ {E}_{li}=\max \left({E}_{l1i},{E}_{l2i}\right) $$
(20)

Therefore, the twist angle and theoretical error can be calculated based on the optimized single-point offset (OSPO) method, as illustrated in Fig. 15 a and b, respectively. According to reference [35], if the theoretical error is approximately less than 5 μm, it can be negligible. As presented in Fig. 15b, the vast majority of theoretical errors based on the OSPO method are less than 5 μm. By comparing Fig. 15b with Fig. 13, theoretical error obtained by using the OSPO method is smaller than that by using the SPO method.

Fig. 15
figure 15

a The twist angle of S-shaped ruled surface obtained using the OSPO method. b The theoretical errors obtained using the OSPO method

Full size image

As presented in Table 2, by comparing with the SPO method, the OSPO method has clear advantages in reducing twist angle and theoretical error.

Table 2 The comparison of twist angle and theoretical error
Full size table

5 Experimental validation

In order to verify the feasibility and effectiveness of the proposed method, an experiment is carried out on the gantry-type five-axis milling machine tools (XKAS2525) and coordinate measuring machine (CMM). The experiment mainly includes two parts: machining and measurement of S-shaped test piece. Fig. 16 shows the flow chart of the verification experiment.

Fig. 16
figure 16

The flow chart of the verification experiment

Full size image

The dimensions and material parameters of the S-shaped test piece are displayed in Table. 1. The ϕ20-mm cylindrical milling cutter is utilized to process the S-shaped test piece. Before the machining test, tool path should be obtained. Tool path planning in this paper is carried out by CAD/CAM software based on the OSPO method. The cutter location file is subsequently generated by the post process module of Pro/E software. Before the cutting test, the NC codes with the iterative compensation methodology presented in reference [3] based on the same compensation strategy are obtained. Then, the flank milling experiment is carried out. The machining site is as illustrated in Fig. 17. Two S-shaped test pieces being machined under the same processing conditions based on the SPO method and the OSPO method are obtained respectively, as shown in Fig. 18. Finally, the contour errors of two test pieces are measured by the coordinate measuring machine (CMM) named PRISMO 125751, which is produced by the German company Zeiss. A ϕ1-mm ruby sphere with stylus length of 40 mm is used in this measurement experiment. Fig. 19 represents the scenes of measurement with a coordinate measuring machine. Three test lines are taken along the Z direction of the S-shaped edge strip, as illustrated in Fig. 20. The distance from L3 to the edge strip top is 5 mm in the Z direction. The interval of the test lines is 12.5 mm. So the recommended measurement points, which located on the L1, L2, and L3, are selected for evaluating the machining performance of five-axis machine tools. The accuracy of the whole surface can be comprehensively examined by the selected three lines. For measuring machining errors, each height is chosen at 30 equidistant discrete points. The referring checkpoint schematic in the CMM system is displayed in Fig. 21.

Fig. 17
figure 17

S-shaped test piece being machined

Full size image
Fig. 18
figure 18

Two test pieces produced using the SPO

Full size image
Fig. 19
figure 19

The detecting scene of coordinate measuring machine and the OSPO methods

Full size image
Fig. 20
figure 20

Position of error detection points

Full size image
Fig. 21
figure 21

Referring checkpoints schematic

Full size image

Before machining, NC machine tools need to warm-up for half an hour. In the measuring process, to reduce the effect of environment temperature on the measurement results, the ambient temperature is kept in the range of 20–22 °C. Moreover, in order to improve the measurement stability, the measurement of kinematic errors was carried out three times, and then the ultimate error values are the average results of three times measured values.

Therefore, the contour errors of 90 points for two S-shaped test pieces at L1, L2, and L3 based on the SPO method and the OSPO method are all obtained, as shown in Fig. 22a–c, respectively. It can be seen that the contour errors of all 90 points obtained by using the OSPO method are much smaller than that obtained by using the SPO method. The negative value denotes overcut error; conversely, the positive value denotes undercut error. The maximum contour error is decreased from 0.264 to 0.135 mm, from 0.322 to 0.138 mm, and from 0.171 to 0.086 mm at L1, L2, and L3 respectively, as illustrated in Fig. 22a–c. In addition, in order to reflect the superiority of the OSPO method more intuitively, the comparison of average contour error for S-shaped test pieces based on the SPO method and the OSPO method is shown in Table 3. It can be clearly seen that the average contour error is reduced from 0.098 to 0.048 mm, from 0.063 to 0.032 mm, and from 0.081 to 0.038 mm at L1, L2, and L3 respectively. In other words, the contour error of S-shaped test pieces based on the OSPO method is reduced by about 51.1%, 46.04%, and 53.1% at L1, L2, and L3 respectively. The average reduction based on the OSPO method is about 50.1%. According to the requirement of United States Patent, the allowable value of machining error of the S-shaped test piece is restricted within ± 0.05 mm, as shown in reference [19]. On the basis of the measurement results, only the test piece 2 can meet the accuracy requirements. Therefore, it is obvious that the proposed OSPO method is precise and effective to reduce the theoretical error of S-shaped test piece. The performance for various kinds of multi-axis machine tools can be accurately exhibited by S-shaped test piece based on the OSPO method.

Fig. 22
figure 22

The contour errors of two S-shaped test pieces obtained using the SPO and OSPO methods

Full size image
Table 3 The comparison of average contour errors for S-shaped test piece
Full size table

6 Conclusion

In this paper, the optimized single-point offset (OSPO) method is proposed to reduce the theoretical error of S-shaped test piece, which has an influence on the precision acceptance of multi-axis machine tools. For the sake of the effective implementation of the method, firstly, the uniform double cubic B-spline surface model is applied to the characterization of the curved surface. By using this model, a three-dimensional model of the S-shaped test piece is established. Then, the theoretical error of S-shaped test piece caused by a twist angle is analyzed. Moreover, to reduce the theoretical error of S-shaped test piece, an optimized tool path offset method, the optimized single-point offset (OSPO) method, was proposed based on traditional single-point offset (SPO) method. In contrast to the SPO method quantitatively, the OSPO method has clear advantages in decreasing theoretical error. Finally, in order to verify the effectiveness and feasibility of the proposed method, a machining and measurement experiment is carried out. Experimental results show the theoretical error of S-shaped test piece based on the OSPO method is reduced by about 51.1%, 46.04%, and 53.1% at L1, L2, and L3 respectively, compared with that based on the SPO method. Furthermore, the average theoretical error of S-shaped test pieces based on the OSPO method is reduced by about 50.1%. Furthermore, most of the theoretical errors based on the OSPO method are approximately less than 5 μm, which can be negligible. Therefore, there is no doubt that this method is feasible and effective enough for reducing the theoretical error. Thus, the performance of multi-axis machine tools can be comprehensively and accurately exhibited by S-shaped test piece based on the proposed method in this paper. The basic idea of the proposed method can be applied to various sculpture surfaces to effectively reduce theoretical errors.