Toolpath planning method with the constraint of cutting force fluctuation in slow tool servo turning for complex curved surface parts

Parts with the complex curved surface are commonly used in high-end equipment. The machining quality of the curved surface parts is important for the property of high-end equipment. With position control of the main spindle, i.e., the C axis, the slow tool servo (STS) turning is a very promising technique to effectively machine complex curved surface parts. However, the turning machining quality is restricted by the cutting force fluctuation in finishing turning. Hence, the STS turning toolpath generation approach with the constraint of the cutting force fluctuation is proposed in this study. As the cutting force is proportionate to the cutting area, the cutting area for turning complex curved surfaces is analyzed and calculated. Then the functional relation between the cutting force fluctuation and the feed rate is derived. Based on the geometrical characteristic of the first revolution of the toolpath, the feed rate is optimized with small cutting force fluctuation to derive the cutter contact points (CCPs) in the cylindrical coordinate system. From the validation experiment results, it can be seen that the cutting force fluctuation, the profile tolerance, and the surface roughness of parts are effectively reduced, and the surface roughness of the part machined with the proposed toolpath planning method is 17.625 nm. Thus, this study provides guidance for toolpath generation with the constraint of the fluctuation of the cutting force in STS turning for a complex curved surface.


Introduction
With the growing demand for high-quality complex curved surface parts, the STS turning operation becomes an alternative way to manufacture crucially mechanical parts for the machine tools, aircraft, optics, etc. [1,2]. With two linear axes, i.e., the X axis and Z axis, and a rotational main spindle, the C axis, STS turning machine is able to fabricate rotationally and non-rotationally symmetric surface parts. Meanwhile, the planning method of the toolpath is vital to STS turning of the complex curved surface. As an important factor of surface quality, the surface roughness of the machining surface has a noticeable influence on the performance of the parts [3,4]. With only two linear axes, i.e., X axis and Z axis, conventional turning machining operations utilize the conservative constant feed rate to fabricate rotational symmetric parts; however, the conservative constant feed rate is neither convenient nor satisfactory in STS turning the complex curved surface, because the geometrical characteristic of parts leads to frequent fluctuation of the cutting force, which is detrimental to the machining quality of the parts and the dynamical performance of the machine tool [5]. Toolpath planning method with consideration of the cutting force change of the complex curved surface STS turning is helpful to diminish the cutting force fluctuation. Moreover, the turning toolpath with small cutting force fluctuation is beneficial to the machining quality. To obtain complex curved surfaces with high machining quality, the STS turning toolpath generation algorithm with the constraint of the cutting force fluctuation for complex curved surfaces is worth researching.
There is much research focusing on the surface generation process of turning to lay the groundwork for the toolpath planning to improve machining quality. [6] investigated the influence of the cutting velocity on the surface generation, then they found that the surface roughness is small with the critical nominal cutting speed. To improve the machining quality, the parameters of the toolpath influencing the surface roughness of machined parts were investigated. [7] proposed an evaluation method taking the process parameters into account to predict the machined surface roughness in single-point diamond turning. Based on the fair prediction of surface roughness, [8] found a conservation law. According to the conservation theory, the process parameters and the tool nose radius of the cutter could be determined according to the assigned surface roughness value. To forecast the machined surface roughness value, [9] proposed a theoretic and empirical coupled approach, and then they employed an intelligent algorithm to search for the best process parameters for the machined surface. Through experimental investigation, [10] found that the feed rate was the principal factor influencing the machined surface roughness, followed by the tool nose radius and cutting speed. Based on the effect of the tool nose radius, [11] constructed a theoretic machining surface roughness model, then they analyzed the effects of the tool nose radius, part material rebound, and the remaining error on the model. To forecast the machined surface roughness in turning of Ti-6Al-4 V, [12] developed a model that involved the cutting velocity, feed rate, and cutting depth as control variables with response surface methodology (RSM), then they proposed that considering dynamic effects of the cutting force gave a more accurate forecast of the surface roughness. To obtain both the machined surface roughness value and the cutting force, [13] applied the RSM to turning of AISI H11 steel with a cubic boron nitride (CBN) tool, then they found that both the feed rate and parts inflexibility have an effect on the surface roughness value. [14] focused on finish dry turning, and they found that the feed rate and cutting velocity obviously influenced the machining quality. When turning with CBN insert, [15] applied the grey relational analysis to optimize the feed rate, cutting depth, and cutting velocity for the machined surface roughness and cutting force. To research the relation between the processing parameters and average surface roughness value during machining the Al/SiC particle composite materials, [16] proposed a surface roughness prediction model, which used the Levenberg-Marquardt backpropagation teaching method. Based on the literature above, it can be found that the effects of the major parameter of the toolpath (cutting velocity and feed rate) and the cutter geometry parameters (tool nose radius and rake angle) on the machined surface roughness were investigated and analyzed through the surface roughness model and cutting experiments, while the impact of dynamics factors (i.e., cutting force and vibration) of processing on the machining quality were rarely analyzed in these researches.
Considering the effect of the cutting force fluctuation on the machining quality, many researchers modeled the cutting force to research the association between the cutting force and the machined surface quality. Allowing for the cutter geometries and the chip formation, [17] presented an analytic cutting force model to enhance the machinability of metal matrix composites. Based on a predictive machining theory, [18] proposed a cutting force model for turning, which adopted the non-equidistant shear zone model and regarded the cutter geometry and processing conditions as the entering information. [19] performed the cutting force forecast by employing Bayesian inference in the cutting force model. A cutting force forecast was performed by employing the posterior probabilities of the model arguments. On the basis of the oblique cutting statement, [20] developed an analytic cutting force forecast model that considers both the edge effects and the size effects. Then, the influence of the processing parameters on the cutting force was researched by some scholars. With the constructed cutting force prediction model, [21] analyzed the influence elements on the fluctuation of the cutting force, then they proposed that the toolpath alongside the little normal vector change direction was beneficial to minimize the cutting force fluctuation in the complicated curved surface parts machining. On the basis of a single-factor experiment, [22] researched the variation laws of the cutting force with the processing parameters in high-speed machining of 300 M steel under cryogenic minimum quantity lubrication conditions. In consideration of the time-variable characteristics of the quasi-intermittent vibration-assisted swing, [23] researched the influence of the processing parameters on the cutting force in the nonequidistant shear zone. Considering the process parameters and material properties in ultrasonic vibration-assisted oblique turning, [24] proposed a prediction model for the instantaneous cutting force. [25] proposed a cutting force model, in which the transient area of the cut was defined by acquiring the beginning time and the stop time of the elliptical vibration texturing. [26] researched the effects of spindle speed, feed rate, and flow rate of cutting fluids on the surface roughness and the cutting force, and the results showed the feed rate had the most effect. To accurately predict the three-direction cutting force in single-point diamond-turning polycrystalline copper, [27] constructed a prediction model considering the grain size of the copper and the stress distribution on the diamond tool. However, most of the research on the dynamic factors is mainly focused on the influence of the cutter geometrics, cutting condition, and processing parameters on the cutting force. In addition, the cutting force forecast model is constructed and evaluated, while the optimal parameter of the turning toolpath with small cutting force fluctuation is rarely studied. Moreover, the effect of geometric feature variation of the complex curved surface on the cutting force is rarely considered, then the varying cutting area in the complex curved surface parts STS turning induces the cutting force fluctuation, which is detrimental to the machined surface quality. Thus, it is sensible to reduce the cutting force fluctuation in the complex curved surface parts STS turning.
This study aims to provide a convenient toolpath planning method to diminish the fluctuation of the cutting force in complex curved surface parts turning. As the cutting force is proportionate to the cutting area of the cutting zone, the cutting force is represented by the cutting area. To reduce cutting force fluctuation in the complex curved surface parts turning, the differential geometrical characteristic of the curved surface parts and the cutting area in turning are analyzed firstly. Based on the analysis of the cutting area in complex curved surface parts turning, in order that cutting area variation is small during machining, the partial derivative of the cutting area about the compound feed rate of two adjacent points on the section curve is deduced when the other process parameters are determined. After the calculation and decomposition of the compound feed rate with the constraint of little fluctuation of the cutting force, the CCPs of the toolpath are calculated recursively based on the first revolution of a toolpath. Then the CLPs of the toolpath are calculated with the tool nose radius compensation. The novelty of this study is that with the consideration of the differential geometrical characteristic variation of the complex curved surface and the cutting force fluctuation in complex curved surface parts turning, the feed rate of the toolpath is optimized for small cutting force fluctuation. Then the toolpath with little cutting force fluctuation for the complex curved surface turning is generated, and high machining quality in verification experiments can be achieved.
The organization of this study is as follows: based on the analysis of the differential geometrical characteristic of the complex curved surface and cutting area of the cutting zone, Sect. 2 derives the functional relation of the cutting area about the compound feed rate. After constructing the functional relation between cutting force fluctuation and feed rate and then optimizing feed rate, Sect. 3 calculates the CCPs and CLPs of the toolpath with the constraint of fluctuation of the cutting force. Section 4 implements the verification experiments, and conclusions are recapitulated in Sect. 5.
2 Analysis of differential geometrical characteristics of complex curved surfaces and calculation of cutting area of the cutting zone.
The cutting area is the area of the cross-section of the cutting layer and the selected plane, which is the rake face of the turning cutter. The cutting area is determined by the tool nose radius, parameters of the toolpath, and the differential geometric characteristic of the surface. In complex curved surface turning, the differential geometric characteristics of the surface part at different points are diverse; thus, it is necessary to analyze the differential geometrical characteristics of complex curved surfaces for cutting area calculation. Meanwhile, when the cutting depth is given, the cutting area of the cutting zone is affected mostly by the compound feed rate, i.e., the distance between adjacent points on the section curve, so the relationship between the cutting area and the compound feed rate of turning is worthy of analysis. Therefore, the differential geometrical characteristic of the complex curved surface and cutting area are analyzed in this section.

Differential geometrical characteristic determination of complex curved surface
To calculate the cutting area of the cutting zone, the required differential geometrical characteristic of the surface is analyzed in this section. In STS turning, as shown in Fig. 1, the part is assembled on and revolved with the main spindle (C axis) of the turning machine tool, and the diamond cutter is installed on the cutter holder which is loaded by the sled of the Z axis. For the cylindrical coordinate system (r, θ, z) is principally adopted in STS turning machining, the transformation from the Cartesian coordinate to cylindrical coordinate and surface parameters for point P = [x, y, z] on surface S(u, v) is derived as follows: where j = 1, 2,⋅⋅⋅, M sum , M sum is the total number of revolutions of the toolpath, u is the parameter in direction of polar radius r, v is the parameter in direction of polar angle θ, θ ∈ [0, 2M sum π], θ is the coordinate value of C axis of the machine tool. r is the absolute value of the coordinate of the X axis of the machine tool, R m = max(r), max() is the function of maximum, and z is the coordinate value of the Z axis of the machine tool.
To analyze the cutting area of the cutting zone, the concave-convex property and the curvature radius ρ of point P on the surface are essential, and then the gaussian curvature K and mean curvature H of the surface need to be worked out first. According to the differential geometry theory, in order to calculate the gaussian curvature K and the mean curvature H, the coefficients E I , F I , and G I of the first fundamental form of the surface and coefficients L II , M II , and N II of the second fundamental form of the surface are derived as follows: where S u is the first-order partial derivative of the surface S(u, v) with regard to the parameter u, and S v is the firstorder partial derivative of the surface S(u, v) with regard to the parameter v, S uu is the second-order partial derivative of S(u, v) with regard to the parameter u. S uv is the mixed second-order partial derivative of S(u, v) with regard to the parameters u and v. S vv is the second-order partial derivative of surface S(u, v) with regard to parameter v.
Then the Gaussian curvature K and the mean curvature H of the surface are respectively calculated as follows: With the Gaussian curvature and the mean curvature of the surface, the principal curvature κ 1 and κ 2 of the surface are respectively calculated as follows: Then the principal directions of principal curvature are respectively calculated as follows: where the coefficients c α and c β are respectively calculated as follows: The S u at the point P is the tangent vector of the section curve between the curved surface and the rake face of the turning cutter, so the angle γ between the tangent vector of the point P and the principal direction Dk 1 is derived as follows: Then the normal curvature κ n of the point P in the direction of the tangent vector is determined by Euler's formula As the cutting engagement situation and the analysis of the cutting area are correlated with the concave and convex shape properties of the surface, it is essential to ascertain the concave-convex property of the point on the section curve. The concave-convex property and the curvature radius ρ of the point P on the section curve are Fig. 1 The configuration of the STS turning machine tool determined by the normal curvature κ n of the surface S(u, v) at the point P as follows: where sgn() is the sign function.
According to the differential geometrical characteristic analysis of the complex curved surface, with the concaveconvex property and the curvature radius ρ of the point P on the surface determined, the cutting area at point P can be analyzed and calculated in the following section.

Analysis and calculation for cutting area of cutting zone in complex curved surface turning
Since the cutting force is proportionate to the cutting area, the cutting force is represented by the cutting area in this study. To evaluate the cutting area of the cutting zone on concave surface and convex surface, as Fig. 2a and  b. showing, set up the local coordinate system X l O l Z l , in which the origin of the coordinate system O l is the circle center of osculation circle of point P 1 on section curve, the Z l axis parallel to the tangent vector of point P 1 and the X l axis is orthogonal to Z l axis. The cutting area of the cutting zone on the plane surface is analyzed in Fig. 2c. P 1 and P 2 are adjacent points on section curve C f , the angle α is ∠P 1 O l P 2 , and ρ 1 and ρ 2 are curvature radiuses of P 1 and P 2 in Fig. 2a and b. The curve C f is the section curve of the complex curved surface and rake face of the turning tool, and the curve C m is the section curve of the uncut surface and rake face of the turning tool. Points A and Q are respectively the centers of adjacent tool nose circles; R is the tool nose radius, and h is the scallop height. Point B and C are respectively the intersection points of curve C m and tool nose circle A, and points M and N are the intersection points of curve C m and tool nose circle Q. Point K is the intersection point of tool nose circle A and tool nose circle Q, and L is the distance of adjacent points on section curve, namely the compound feed rate in X and Z directions of turning machine. The cutting zone is composed of the arcs of MP 2 K, KB, and MB on the concave surface and the convex surface, or the arcs of MP 2 K, KB, and segment MB on the plane surface, as the yellow zone shows. The area of sector MQN is denoted as S MQN , the area of triangle △MNQ is denoted as S tri , the area of sector , and the area of BNK portion is denoted as the area of triangle △BNK S tri . Thus, the cutting area of the cutting zone is calculated as follows: where " ± " is replaced by " + " when sgn(κ n ) = − 1 and replaced by "-" when sgn(κ n ) = 1.
The area of the sector MQN is calculated as follows: where β = ∠MQN. The area of triangle △MNQ is calculated as follows: According the double angle formula as Eq. (14), the area of triangle △MNQ is calculated as Eq. (15).
With the concave-convex property and the curvature radius ρ of the point P on the section curve are determined in Sect. 2.1, in Fig. 2a and b., there is ρ = ρ 1 = ρ 2 in the local zone, ∠MO l N is approximated to α for the cutting depth a p is much less than the cutter nose radius R; thus, the area of the sector MO l N is calculated as follows: in which, " ± " is replaced by " + " when sgn(κ n ) = − 1 and replaced by "-" when sgn(κ n ) = 1. The area of triangle △ MO l N is calculated as follows: in which, sinα = L/ρ, " ± " is replaced by " + " when sgn(κ n ) = − 1 and replaced by "-" when sgn(κ n ) = 1.
According to the similarity of triangle △ BNK and triangle △P 2 P 1 K, there is Eq. (18), S tri in the cutting zone of a plane surface is calculated as Eq. (19): Then according to the similarity of triangle △ BNK and triangle △ P 2 P 1 K, the area of triangle △BNK in the cutting zone of the concave surface and the convex surface is calculated as follows: Therefore, the cutting area of the cutting zone is calculated as follows: where " ± " is replaced by " + " when sgn(κ n ) = − 1 and replaced by "-" when sgn(κ n ) = 1.
According to the double angle formula as Eq. (22), the Eq. (23) is derived to calculate cos 2 . To avoid machining interference, the tool nose radius R is less than ρ; in addition, the L is less than ρ, for L is in micron-scale in finishing turning. Then α is less than π, thus the Eq. (21) is derived as Eq. (24). With the Eq. (24), the cutting area of the cutting zone is conveniently derived through the tool nose radius, the cutting parameters, and the geometric characteristics of the surface.
As the cutting area of the cutting zone is calculated as Eq. (24), regard it as the functional relationship between the cutting area and the compound feed rate. Since the cutting force is proportional to the cutting area, the functional relation between the feed rate and the cutting force fluctuation is constructed in the next section.

STS turning toolpath planning with the constraint of small cutting force fluctuation
With the differential geometrical characteristic and the cutting area are analyzed and determined, the toolpath is derived in this section. As the partial derivative of the cutting area about the compound feed rate represents the rate of variation of the cutting force about the compound feed rate, this section constructs the functional relation between the feed rate and the cutting force fluctuation. Furthermore, the feed rate is determined under the constraint of a small fluctuation of cutting force in different geometry feature of a surface. With conventional equal-angle scattered CCPs of the first revolution of the toolpath, the CCPs on the section curve are derived recursively with the feed rate along the X axis serving as polar radius variation of cylindrical coordinate. Then the CLPs of the toolpath are calculated with tool nose radius compensation to generate the complex curved surface STS turning toolpath with the constraint of small fluctuation of the cutting force.

Construction of functional relation between feed rate and cutting force fluctuation
Regarding the cutting area S as the function of compound feed rate L, the partial derivative of the cutting area with regard to the compound feed rate represents the rate of variation of the cutting area with regard to the compound feed rate: therefore, the partial derivative of cutting area with regard to compound feed rate is proportional to the fluctuation of cutting force.
To construct the connection between the feed rate and the fluctuation of cutting force, this section calculates the partial derivative of the cutting area with regard to the compound feed rate. For the sake of simplicity, the point P 1 in the illustration of the cutting area is denoted as point P, and then the partial derivative of the cutting area with regard to the compound feed rate is calculated in the following three situations according to the differential geometrical characteristic of point P on the section curve.
(1) When the local surface around the point P is a convex surface, according to Eq. (9) and (24), the partial derivative of the cutting area S with respect to the compound feed rate L of adjacent points on the section curve can be derived as follows: (2) When the local surface around the point P is a concave surface, according to Eq. (9) and (24), the partial derivative of the cutting area with respect to the compound feed rate of adjacent points on the section curve can be derived as follows: (3) When the local surface around the point P is the plane surface, according to Eq. (9) and (24), the partial derivative of the cutting area with respect to the compound feed rate of adjacent points on the section curve can be derived as follows:

Method of CCPs calculation with the constraint of small cutting force fluctuation
In order to derive the CCPs of the toolpath, firstly concerning the first revolution of the toolpath, the compound feed rate with the constraint of scallop height is adopted, and the compound feed rate serves as the polar radius variation of the first revolution of the toolpath. The functional relationship between the compound feed rate L and the scallopheight h is convenient to reach according to the differential geometrical characteristic of the surface as follows: Since the partial derivative of the cutting area with respect to the compound feed rate is proportionate to the cutting force fluctuation, to solve the optimal feed rate with a small partial derivative, the partial derivative of the cutting area with regard to the compound feed rate should be minimum; thus, the optimal feed rate of toolpath is respectively derived according to the differential geometrical characteristic of the point P as follows: (1) When the local surface around the point P is a convex surface, with Eq. (25) assigned to zero, the compound feed rate L of adjacent points on the section curve is calculated as follows: (2) When the local surface around the point P is a concave surface, according to Eq. (26), the partial derivative of the cutting area with regard to the compound feed rate is always less than zero, and it denotes that there is a negative correlation between the rate of change of cutting area and compound feed rate. According to the ab ; thus, the optimal compound feed rate of adjacent points on the section curve is determined by solving the nonlinear equation Eq. (30): h( −a p ) 2 into Eq. (30), Eq. (31) is derived as follows: Solving the nonlinear equation Eq. (31), the real root is the desirable value of t, and then the compound feed rate L of adjacent points on the section curve is calculated as follows: (3) When the local surface around point P is a plane surface, according to Eq. (27), the partial derivative of the cutting area with respect to the compound feed rate is a negative constant. It denotes that there is a fixed negative rate of variation of the cutting area with regard to the compound feed rate. For the rate of variation of the cutting area with regard to the compound, the feed rate is fixed, according to Eq. (28), the compound feed rate of adjacent points on the section curve is determined as follows: With the derived Eqs. (29) to (33), the compound feed rate of adjacent points could be calculated. Specifically, the parameters of a p , h, and R in the equations are given, and the concave-convex property and the curvature radius ρ of the point on the section curve are determined by the analysis of differential geometrical characteristics. To derive the polar radius of a point on the toolpath, the feed rate along the X axis of the turning machine needs to be calculated by decomposing the compound feed rate to the feed rate along the X and Z axes of the turning machine tool. Denoting the direction vector of the Z axis as dir Z = [0, 0, −1] T , the angle φ between the tangent vector in the feed direction of point P and dir Z is calculated as Eq. (34). Then the feed rate along the X and Z axes of the turning machine is respectively decomposed as f X and f Z through Eq. (35) and (36): With the calculated compound feed rate in Eq. (28) serving as the polar radius variation f X,N num ,1 of the first revolution, N num is the number of equal-angle scattered points in one revolution of the toolpath. Scatter the first revolution of toolpath to points by the equal polar angle in polar coordinate as shown in Fig. 3, with the feed rate of the X axis at the i-th point in j-th revolution P i,j served as the polar radius variation f X,i,j , the polar radius of succeeding CCP P i,j+1 on the section curve with same polar angle is calculated as follows: where r i,j is the polar radius of the i-th point in the j-th revolution of the toolpath, i = 1,2,⋅⋅⋅, N num .
The polar radius of CCP P i,j+1 serves as the coordinate of the X axis, and the coordinate of the C axis θ of point P i,j+1 is derived as follows: in which, θ 0 is the scattered equal-angle value of the toolpath, θ 0 = 2π/N num .
Then the CCPs on the section curve of each polar angle can be derived recursively. With the differential geometrical characteristic of point P i,j+1 derived according Sect. 2.1, the optimal feed rate of the toolpath can be calculated; thus, the CCPs of the toolpath are derived.

CLPs calculation with tool nose radius compensation
The numerical control code for machining is programmed through the cutter location point (CLP), which corresponds one-to-one with the cutter contact point (CCP). With the CCP P CC = S(u i,j , v i,j ) of the toolpath calculated, the CLP P CP of the toolpath is derived with tool nose radius compensation as follows: where n tool is the unit normal vector of rake face P rake as shown in Fig. 4 , n p is the projection vector of surface normal vector n on the rake face.
With the CLPs of the toolpath calculated, the toolpath under the constraint of the fluctuation of cutting force for complex curved surface STS turning is generated. Then the verification and comparison experiments are conducted in Sect. 4.

Verification experiments and results
In this section, verification experiments are implemented to validate the practicability and effectiveness of the proposed turning toolpath planning method for improving the machining quality. The contrast tests are respectively conducted with the constant feed rate toolpath and the constant scallopheight toolpath.

Compared with constant feed rate toolpath machining
To demonstrate the effectiveness of the proposed toolpath generation method, the contrastive verification experiment is first implemented on a rotary cosine surface part, as shown in Fig. 5. The material of the part is aluminum alloy 6061, and the radius of the part is 30 mm. The equation of the generatrix in the XOZ plane of the machine tool is  To generate the toolpath with the constraint of the fluctuation of cutting force for a rotary cosine surface part, at first, the differential geometrical characteristics of the point on the generatrix are analyzed with the proposed method in Sect. 2.1. Then the cutting area is analyzed and calculated according to the proposed algorithm in Sect. 2.2. The optimal feed rate is derived according to the constructed functional relation between the feed rate and the fluctuation of cutting force in Sect. 3.1. The CCPs and CLPs of the toolpath are respectively calculated with the method in Sects. 3.2 and 3.3. The toolpath of the rotary cosine surface part generated by utilizing the method proposed in this study as shown in Fig. 6.
The finishing turning experiment details are listed as follows: the experiments are conducted on the turning machine (41) X = 4cos(0.1Z ) − 0.1Z + 26 Fig. 6 The turning toolpath of the rotary cosine surface part Fig. 7 Finishing machining equipment, measurement machine tool, and machining parts. a Finishing machining equipment, b measurement machine tool, c part machined with toolpath planned by the proposed method, and d part machined with constant feed rate toolpath tool CAK3665, which is made by SMTCL. The positioning accuracy of the X axis is 0.03 mm, and the positioning accuracy of the Z axis is 0.04 mm. The repeatability positioning accuracy of the X axis is 0.012 mm, and the repeatability positioning accuracy of the Z axis is 0.016 mm. The parameters of the polycrystalline diamond cutters are the same in this experiment. The polycrystalline diamond cutter with a circular insert has a tool nose radius of 0.8 mm, a rake angle of 0°, a clearance angle of 7°, and included angle of 145°. The included angle is the central angle of the circular insert available for cutting. The cutters for the contrast experiments are both new tools, so the tool wear is ignored in the experiments. The cutting depth a p is 0.4 mm, the scallop height is 0.1 μm, the spindle speed n s is 200 r/min (revolutions per minute), and the constant feed rate along the Z axis of the turning machine tool of the controlled experiment is 0.013 mm/r (millimeter per revolution), which takes equivalent machining time compared with the toolpath planned by the proposed method in this study. The piezoelectric three-component cutting force measuring system is YDC-III89C, and the sampling frequency of the measuring system is 1 kHz.
The experimental equipment is shown in Fig. 7a. For getting an assessment of the surface quality of the machining surface, the surface roughness is measured by the Talyrond Hobson surface roughness and profile tester as shown in Fig. 7b. The part machined with the toolpath planned by the proposed method is shown in Fig. 7c, and the part machined with a constant feed rate toolpath is shown in Fig. 7d. In order to ensure the credibility of the experimental result, repeated experiments are conducted.
To comprehensively evaluate the cutting force, the resultant cutting force of the measured tangential, radial, and axial components of the cutting force is shown in Fig. 8. The resultant cutting force in processing with a constant feed rate toolpath is shown in Fig. 8a; the resultant cutting force in processing  Fig. 8b; the contrast of resultant cutting force in the local region is shown in Fig. 8c. The standard deviation of resultant cutting force in processing with a constant feed rate toolpath is 8.4 N, which evaluates the fluctuation, while the resultant cutting force standard deviation of processing with the toolpath generated by utilizing the method proposed in this study is 7.5 N, reduced by 10.71%. It demonstrates that the cutting force fluctuation of processing with toolpath generated by utilizing the method proposed in this study is decreased effectively.
The measured surface roughness of the part machined with the toolpath planned by the proposed algorithm is shown in Fig. 9a, and the measured surface roughness of the part machined with a constant feed rate toolpath is shown in Fig. 9b. The average surface roughness Ra of the part machined with a constant feed rate toolpath is 0.3424 μm. The average surface roughness Ra of the part machined with the toolpath planned by the proposed algorithm is 0.2585 μm, which is 24.50% less than that of the part machined by the constant feed rate toolpath.
The profile of the machined part is measured with the ZEISS PRISMO navigator measuring machine tool, whose measurement error is not more than 1 μm. The measuring process is shown in Fig. 10. The measurement points of the rotary cosine surface part machined with the proposed method are shown in Fig. 11. The profile tolerance of the part machined with the constant feed rate toolpath is 37.1 μm. The profile tolerance of the part machined with the toolpath planned by the proposed method is 32.9 μm, which is 11.32% less than that of the part machined by the constant feed rate toolpath. The experimental results of turning and machining the rotary cosine surface parts are listed in Fig. 12 and Table 1.

Compared with constant scallop-height toolpath machining
To confirm the usefulness of the proposed toolpath generation method further, the second comparison experiment is implemented with the constant scallop-height machining on the STS machine tool. The verification experiment is conducted on an optical off-axis sphere surface part whose material is aluminum alloy 2A12, and the radius R c of the part is 20 mm, as shown in Fig. 13. The equation of the off-axis sphere surface in the Cartesian coordinate system is in which, the radius of sphere R s is 200 mm, and the off-axis distance H d is 30 mm.
To generate the toolpath under the restriction of fluctuation of the cutting force for the off-axis sphere surface part, The finishing turning machining experiment details are listed as follows: the experiments are implemented on the STS turning machine tool. The parameters of the single crystal diamond cutters are the same in the comparison experiments, and the turning cutter is a single crystal diamond cutter with a circular radius of 2 mm, rake angle of 0°, clearance angle of 15°, and included angle of 120°.   The cutters for the comparison experiments are both new tools, so the tool wear is ignored in the experiments. The cutting depth a p is 0.004 mm; the spindle speed n s is 50 r/min, and the constant scallop height of the controlled experiment is 10 nm.
The turning machining toolpath is generated with the proposed method in this study, compared with the constant scallop-height toolpath. The experimental equipment is shown in Fig. 15a. Since the cutting force fluctuation is effectively decreased in the former verification experiment in Sect. 4.1, and because the structure of the tool turret of the STS turning machine is different from that of the turning machine in the former verification experiment in Sect. 4.1, the cutting force is not measured in this experiment. To ensure the credibility of the turning experiment, repeated experiments are conducted. The part machined with a toolpath generated with the proposed method is shown in Fig. 15b, and the part machined with a constant scallopheight toolpath is shown in Fig. 15c.
The surface roughness is measured by S neox, which is a three-dimensional optical profiler. The measured surface roughness of the part machined with the toolpath generated with the proposed method is shown in Fig. 16a, and the measured surface roughness of the part machined with constant scallop-height toolpath is shown in Fig. 16b. The average surface roughness Sa of the part machined with a constant scallop-height toolpath is 21.178 nm. The average surface roughness Sa of the part machined with the toolpath planned by the proposed method is 17.625 nm, which is 16.78% less than that of the part machined by the constant scallop-height toolpath.
The profile of the machined part is measured with the ZEISS PRISMO navigator measuring machine tool, whose measurement error is not more than 1 μm. The measurement points of the off-axis sphere surface part machined with the proposed method are shown in Fig. 17. The profile tolerance of the part machined with the constant scallop-height toolpath is 12.6 μm (Fig. 18). The profile tolerance of the part machined with the toolpath planned by the proposed method is 10.9 μm, which is 13.49% less than that of the part machined by the constant scallop-height toolpath. The experimental results of turning the off-axis sphere surface parts are listed in Table 2.

Conclusions
In this study, an STS turning toolpath generation method under the restriction of fluctuation of cutting force for the complex curved surface part is proposed. Differential geometrical characteristic of the complex curved surfaces is first analyzed, and then the functional relationship of the cutting area and compound feed rate is constructed. To construct the functional relation between the STS turning cutting force fluctuation and the compound feed rate, the cutting force is represented by the cutting area, and then the partial derivative of the cutting area with regard to the feed rate is derived. The CCPs and CLPs of the toolpath under the constraint of little fluctuation of cutting force can be generated  (1) Based on the analysis of the differential geometrical characteristic of complex curved surface and calculation of the cutting area, the functional relation of the cutting  area with regard to the feed rate is constructed in consideration of the process parameters and the geometric feature variation of complex curved surface parts. (2) As the cutting force is proportional to the cutting area, the fluctuation of the cutting force is represented by the partial derivative of the cutting area with respect to the compound feed rate of adjacent points on the section curve between the curved surface and the rake face of the turning cutter. Then the feed rate of the toolpath with a small fluctuation of cutting force is calculated. Thereafter, the CCPs and the CLPs of the toolpath are derived on the basis of the points in the first revolution of the toolpath. (3) Verification experiments show that with the proposed toolpath planning algorithm, the cutting force fluctuation is reduced by 10.71%, the profile tolerance is diminished by 11.32%, meanwhile, the machining surface roughness is reduced by 24.50% compared with that of the constant feed rate toolpath. Furthermore, in precision STS turning, the profile tolerance of the part is diminished by 13.49% compared with that of the constant scallop-height toolpath, and the machining surface roughness of the proposed toolpath planning method is 17.625 nm, which is reduced by 16.78% compared with that of the constant scallop-height toolpath. It is confirmed that with the proposed toolpath planning algorithm, the fluctuation of cutting force, the profile tolerance, and the machining surface roughness of the parts can be reduced effectively; hence, the machining quality is improved.
This research provides a method to plan the STS turning toolpath under the restriction of a small fluctuation of the cutting force. The achievements are meaningful for the enhancement of the machining quality of the parts, which presents guidance for the STS turning of complex curved surface parts.

Declarations
Ethics approval Not applicable.
Consent to participate All authors agree to participate.

Consent for publication
All authors agree to publish.

Conflict of interest
The authors declare no competing interests. Fig. 18 The profile tolerance of the machined off-axis sphere surface parts