*Corr. Author’s Address: Jilin University, School of Mechanical and Aerospace Engineering, China, jzhao@jlu.edu.cn 343
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351 Received for review: 2021-03-20
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-05-18
DOI:10.5545/sv-jme.2021.7172 Original Scientific Paper Accepted for publication: 2021-06-17
Ultra-Precision Single-Point Diamond Turning  
of a Complex Sinusoidal Mesh Surface  
Using Machining Accuracy Active Control 
Ning, P.X. – Zhao, J. – Ji, S.J. – Li, J.J. – Dai, H.D.
Peixing Ning
1
 – Ji Zhao
1,2,*
 – Shijun Ji
1
 – Jingjin Li
1
 – Handa Dai
1
1
Jilin University, School of Mechanical and Aerospace Engineering, China 
2
Northeastern University, School of Mechanical Engineering and Automation, China
Single-point diamond turning (SPDT) assisted with slow tool servo (STS) is the most commonly utilized technique in the fabrication of optical 
modules. However, the tool path significantly affects the quality of the machined surface. In order to realize the determined machining 
accuracy effectively, a tool path generation (TPG) method based on machining accuracy active control (MAAC) is presented. The relationship 
between tool path and machining error is studied. Corner radius compensation (CRC) and the calculation of chord error and residual error are 
detailed. Finally, the effectiveness of the proposed approach is verified through a machining error simulation and a cutting experiment of a 
complex sinusoidal mesh surface fabrication.
Keywords: machining accuracy active control (MAAC), machining error prediction, complex sinusoidal mesh surface, single-point 
diamond turning (SPDT)
Highlights
• 	 A tool path generation method based on active control machining error is proposed; the relationship between tool path 
generation parameters and machining error, including chord error and residual error, is studied. 
• 	 The analytical geometry method and tool contact point discretization method are used to realize tool path planning. 
• 	 The presented method can predict machining error and achieve the surface accuracy control before actual processing.
0  INTRODUCTION
Single point diamond turning (SPDT) is an 
economical and efficient fabrication technology that 
can achieve an optical finish without subsequent 
machining [1] and [2]. At present, complex surfaces 
are widely used in various areas [3] and [4]. Therefore, 
scholars have conducted many studies on the 
ultra-precision machining technology of complex 
optical surfaces. Kong et al. [5] presented a hybrid 
tool servo (HTS) process and proposed a tool path 
generation (TPG) method. A microstructure array 
compound freeform surface machining experiment 
verified the validity of the proposed theory. Zhang 
et al. [6] put forward a termed toroid-surface-based 
slow tool servo (STS) turning method to generate 
discontinuously structured micro-lens arrays. Tian et 
al. [7] developed a novel fast tool servo system and 
experimented with rear-view mirrors fabrication 
using that system. Based on the principle of automatic 
dynamics analysis of mechanical systems, Khagani 
and Cheng [8] introduced an innovative approach for 
TPG. In their research, the Newton-Raphson method 
was used to generate the tool paths of very complex 
freeform surfaces; the study revealed that the time 
step size is very important in that method [8]. Li et 
al. [9] investigated a systematic approach for TPG and 
a theory for surface topography simulation in SPDT. 
A sinusoidal grid and micro-lens array sample were 
machined and measured to validate the effectiveness 
of the proposed theoretical research. Chen et al. [10] 
put forward a triangle rotary method; the results of 
the simulations and experiments showed that the 
presented approach was very feasible for position-
velocity-time (PVT) interpolation. The presented 
method can decrease the interpolation error. Wei et al. 
[11] set up a progressive addition lens design model 
and proposed an optimized TPG method for diamond 
turning of the optical freeform surface. Ji et al. [12] 
designed and machined a compound freeform surface 
using an optimized TPG approach. Fang et al. [13] 
studied a cylindrical coordinate machining method for 
freeform surfaces fabrication, in which a typical non-
uniform rational B-spline (NURBS) was utilized to fit 
the feature points. They also carried out a machining 
experiment of a compound eye structure surface to 
prove the theory. The above research studies mainly 
focused on TPG, corner radius compensation (CRC) 
and machined surface topography analysis. However, 
few studies have been conducted on the basis of TPG 
to ensure machining accuracy. In this study, a TPG 
method based on machining accuracy active control 
(MAAC) is proposed, and the relationship between 
TPG parameters and machining error, including chord 
error and residual error, is studied.

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
344 Ning, P.X. – Zhao, J. – Ji, S.J. – Li, J.J. – Dai, H.D.
This paper proposed a TPG method by MAAC 
for complex surface SPDT assisted by STS. In Section 
2, the presented method and CRC are described 
minutely. The relationship between tool path and 
machining errors is studied. An experiment for a 
complex sinusoidal mesh surface fabrication utilizing 
the proposed method is carried out in Section 3. 
Finally, Section 4 summarizes the paper.
1  TPG BASED ON MAAC
1.1  Machining Accuracy Active Control Method
In the process of complex mirror-like surface 
machining, TPG is the first and the most critical step, 
which has a visible impact on surface topography 
and surface finish of the machined workpiece. The 
SPDT technology is based on spindle rotary motion 
and linear reciprocating motion. The spindle rotation 
and the Z-axis reciprocating motion produce the 
chord error in the direction of cutting motion, and the 
X-axis linear motion produces the residual error in the 
direction of feed motion. 
Considering the above two principle errors, 
the TPG method based on MAAC can be used as 
the selection basis of cutting contact points (CCPs). 
Thus, the tool path for SPDT of the complex surface 
that meets the requirements of machining errors can 
be obtained. The method is generally divided into the 
following two steps: 
Firstly, the feed f should be calculated according 
to the requirement of residual error. Because there 
is a certain interval between two adjacent circles of 
the tool path, and the shape of the cutting edge is a 
circular arc, the cutting residue will be produced in 
the direction of cutting motion during the machining 
process. To calculate the residual error, the section 
curve in radial direction z = f (r) should be obtained, 
and the curvature radius R
i
 at CCP P
i
(r
i
, z
i
) should be 
calculated by Eq. (1):
 R
K
fr
fr
i
i
i
i

 1 1
23 2
(( () ))
()
,
/
'
''
 (1)
where K
i
 is the curvature at each CCP, f '(r
i
) is the 
first derivative of the section curve, f ''(r
i
) is the 
second derivative of the section curve. According to 
the difference of curvature, the calculation of cutting 
residual height can be divided into three forms [14]; 
the principle is indicated in Fig. 1.
Fig. 1.  Schematic diagram of residual error calculation

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
345 Ultra-Precision Single-Point Diamond Turning of a Complex Sinusoidal Mesh Surface Using Machining Accuracy Active Control   
When the requirement of residual error is Δ
res
, the 
feed f
i
 at CCP P
i
(r
i
, z
i
) can be obtained through Eqs. 
(2) and (3):
fr R
f
Rr R
re resi
resr es resr es
1
2
22 0
222
 

   
s
i
()
() ()

 i i
resi
i
resr es ir es resi
r
R
R
f
Rr R
 



    
2
0
222
3


()
() () 2 2
0
r
R
R
resi
i















()
, (2)
 ff ff
i
= min{ ,,}.
123
 (3)
Secondly, the TPG could be completed on the 
basis of the requirement of chord error. In the cutting 
process, the tool path between two adjacent CCPs is 
a straight line rather than an ideal curve. The chord 
error is the linear error between a realistic tool 
path and an ideal curve, as presented in Fig. 2. The 
analytical mathematical optimization TPG method 
can be used to realize active control for chord error: 
The tool path can be expressed as z = f ( φ). Because the 
distance between two adjacent CCPs is very small, the 
tool path between two points can be approximated as 
a circular arc. Therefore, the chord error Δ
cho
 ≈  δ
chord
. 
According to the geometric relationship of incipient 
point P
j
 and required chord error Δ
cho
, the next CCP 
P
j+1
 could be derived from Eq. (4):
  

choc hordj j
jj
R
PP
 =R
2
1
2
2
() . (4)
Repeating the above process with P
j+1
 as the 
initial point, the TPG can be finally achieved based on 
active control for chord error.
1.2  Corner Radius Compensation (CRC)
In the SPDT process, if the vertex of the cutting edge 
always moves along the CCPs, overcut will occur 
when curvature is not zero, and a large machining 
error will be leaded. To make the cutting edge always 
tangent to the section curve in the radial direction of 
the proposed surface in the machining process, the 
CRC is needed. Unlike the process of milling, the 
CCPs needs to be compensated in three directions of 
space. Since there is no Y direction motion in SPDT, 
the CRC only needs to consider the compensation in 
the XZ plane. The principle of CRC is detailed in Fig. 
3.
Fig. 3.  Schematic diagram of CRC
Fig. 2.  Schematic diagram of chord error calculation

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
346 Ning, P.X. – Zhao, J. – Ji, S.J. – Li, J.J. – Dai, H.D.
The black line is the original position without 
CRC. At this point, the shaded area indicates that 
overcut occurred. To avoid overcut and keep the 
cutting point position constant, the trajectory of the 
CCPs P
i
(r
i
, z
i
) offsets a length of corner radius along 
the normal vector direction n
r
i
 and the CLPs P
i
(r
i
, z
i
) 
is obtained, as written in Eq. (5):
 
rr r
fr
fr
zz r
fr
ii
i
i
ii
i
'
'
''
'
''













()
()
()
.
1
1
1
2
2
 (5)
Fig. 4.  Form chart of complex sinusoidal mesh surface machining
2  MACHINING EXPERIMENT AND DISCUSSION
To prove the feasibility of the proposed method, a 
machining experiment of a complex sinusoidal mesh 
surface is conducted. Fig. 4 presents the flowchart of 
SPDT process for the surface machining.
2.1  Complex Sinusoidal Mesh Surface Model Establishing
The designed complex sinusoidal mesh surface is 
combined with umbrella surface and radial sinusoidal 
surface with a gradual amplitude gradient. Therefore, 
the mathematical model can be established as Eq. (6):
 
Z  




















h
R
r
R
l
h
R
r
h
R
r
w
w
w
w
11
2
2
2
2
sin
sin





  






h
R
r
w
2
, (6)
where     
h
R
r
R
l
h
R
r
w
w
w
11
2
2














sin



     expresses 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
347 Ultra-Precision Single-Point Diamond Turning of a Complex Sinusoidal Mesh Surface Using Machining Accuracy Active Control   
umbrella surface and 
h
R
r
h
R
r
ww
22
2














sin



indicates complex radial sinusoidal surface; h
1
 and 
h
2
 are amplitudes of the two surfaces; l and ω are 
the number of sinusoids with complete periods in 
circumferential and radial directions, respectively; r is 
polar radius, R
w
 is radius of the workpiece, φ is polar 
angle.
2.2  TPG and Machining Error Prediction
The designed complex sinusoidal mesh surface is 
modelled as Eq. (10) with the maximum amplitude 
h
max
 = 0.2 mmm, h
1
 = h
2
 = h
max
/4, the number of 
sinusoids with complete periods in a circumferential 
direction is l = 8, the number of sinusoids with 
complete periods in radius direction is ω = 32/15, the 
radius of workpiece R
w
 = 8 mm, as shown in Fig. 5. 
The Δ
res
 is set as 0.5 μm and Δ
cho
 is also set as 0.5 μm. 
The TPG method based on MAAC is used for tool 
path planning of the designed surface. Fig. 6 presents 
the planning result.
Fig. 5.  Complex sinusoidal mesh surface model
Fig. 6.  Tool path planning of complex sinusoidal mesh  
surface machining
Reversing the calculation process of TPG applied 
above, the prediction of machining errors can be 
realized. The prediction consequence indicated that 
the errors are all maintained within 0.5 μm as shown 
in Fig. 7.
a) 
b) 
Fig. 7.  Machining error prediction; a) chord error prediction,  
and b) residual error prediction
The CLPs after CRC, which can be directly 
applied to computerized numerical control (CNC) 
machine for complex sinusoidal mesh surface 
fabrication, is depicted in Fig. 8:
Fig. 8.  CLPs and CCPs trajectory of complex sinusoidal mesh 
surface machining
2.3  Machining and Measurement Experiments
The processing experiment is conducted by using a 
Nanoform 250SPDT machine. The workpiece is made 
of Al6061. The cutting parameters are summarized in 
Table 1. The machine tool system and successfully 
machined sample are exhibited in Fig. 9. 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
348 Ning, P.X. – Zhao, J. – Ji, S.J. – Li, J.J. – Dai, H.D.
Table 1.  Cutting parameters
Parameters Value
Tool rake angle γ 0°
Tool clearance angle α 10°
Tool included angle ε 120°
Corner radius r
ε
0.496 mm
Depth of cut a
p
5 μm
Feed f 0.0402 mm
Average cutting speed v
c
166.72 mm/min
a) 
b) 
Fig. 9.  The machining experiment; a) machine tool system,  
and b) the machined complex sinusoidal mesh surface sample
To verify the machining quality, the machined 
surface is measured using a newview 9000 white 
light interferometer profiler Zygo, as shown in Fig. 
10. The measurement conditions are as follows: 
the enlargement factor of zoom lens was set as 1.0× 
(standard). The enlargement factor of the objective 
lens was set as 50×. The filter tray was set as Standard 
F1, which can be used for most surfaces measurement. 
The bandwidth of the filter tray Standard F1 is 125 
nm, and the centre wavelength is 550 nm. The 
field of view in object-space was set as 0.87 mm × 
0.87 mm. The measurement adopts the method of 
distributing points at equal distances. The number of 
distributing points in the X and Y directions are all 
1000. Due to the limitation of the measurement vision 
field, eight areas on different radii of the machined 
surface were randomly selected for measurement, 
including peak areas, valley areas and interim areas. 
The measurement results and the obtained machining 
errors are presented in Fig. 11. Figs. 11 (a1) to (h1) are 
reconstructed surfaces using the measurement data, 
and Figs. 11 (a2) to (h2) are machining errors after 
removing the form component of the machined surface 
from the original measurement data. Because the 
residual error and the chord error are coupled, which 
is inconvenient to measure and analyse separately, 
the machining quality can be evaluated through the 
peak valley (PV) value of the machined surface. PV 
represents the maximum peak-valley deviation of the 
form error of the machined surface. The residual error 
and the chord error could be negative. So the predicted 
PV can be calculated through Eq. (7):
 PV m      () () .
resc ho resc ho
2 (7)
From Fig. 11, excluding the random error, 
the overall deviation (actual PV) of the machined 
complex sinusoidal mesh surface is about 2.459 μm, 
which is a little bit larger than the predetermined PV 
2 μm, because there are some other uncontrollable 
factors such as measurement error, machine tool error, 
tool wear and cutting temperature. Therefore, it also 
can be demonstrated from the experiment result that 
the TPG by MAAC is usable.
3  CONCLUSIONS
In this paper, a TPG method based on MAAC is 
presented by studying the mapping relationship 
between tool path and processing errors. The 
simulation and experiment of a complex sinusoidal 
mesh surface machining attest to the validity of the 
method. The conclusions are drawn as follows:
1. According to the requisition of residual error and 
chord error, the tool path that meets the machining 
Fig. 10.  The measurement experiment

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
349 Ultra-Precision Single-Point Diamond Turning of a Complex Sinusoidal Mesh Surface Using Machining Accuracy Active Control   
a1)           a2) 
b1)           b2) 
c1)            c2) 
d1)            d2) 
e1)            e2) 

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
350 Ning, P.X. – Zhao, J. – Ji, S.J. – Li, J.J. – Dai, H.D.
4  ACKNOWLEDGEMENTS
This work is supported by Key R&D Projects of the 
Ministry of Science and Technology of China (Grant 
Nos. 2017YFA0701200 and 2018YFB1107600) 
National Natural Science Foundation of China (Grant 
No 51775237), Graduate Innovation Fund of Jilin 
University (Grant No. 101832020CX122).
5  REFERENCES
[1] Zhang, S.J., Zhou, Y.P., Zhang, H.J., Xiong, Z.W., To, S. (2019). 
Advances in ultra-precision machining of micro-structured 
functional surfaces and their typical applications. International 
Journal of Machine Tools and Manufacture, vol. 142, p. 16-41, 
DOI:10.1016/j.ijmachtools.2019.04.009.
[2] He, C.L., Zong, W.J., Xue, C.X., Sun, T. (2018). An accurate 3D 
surface topography model for single-point diamond turning. 
accuracy demand can be derived by reverse 
application of the error calculation process.
2. Using the proposed method, the TPG and CRC for 
a complex sinusoidal mesh surface machining are 
carried out systematically. The machining error 
simulation manifests that the generated tool path 
can meet the machining accuracy requirement.
3. Machining and measurement experiment results 
show that the PV of the machined complex 
sinusoidal mesh surface is not significantly 
different from the predetermined PV, which 
attest to the effectiveness of the TPG method by 
MAAC.
f1)            f2) 
g1)            g2) 
h1)            h2) 
Fig. 11.  Measurement results and machining errors; (a1 to h1) reconstructed surfaces, and (a2 to h2) machining errors

Strojniški vestnik - Journal of Mechanical Engineering 67(2021)7-8, 343-351
351 Ultra-Precision Single-Point Diamond Turning of a Complex Sinusoidal Mesh Surface Using Machining Accuracy Active Control   
International Journal of Machine Tools and Manufacture, vol. 
134, p. 42-68, DOI:10.1016/j.ijmachtools.2018.07.004.
[3] Cai, H.B., Shi, G.Q. (2019). Tool path generation for multi-
degree-of-freedom fast tool servo diamond turning of optical 
freeform surfaces. Experimental Techniques, vol. 43, p. 561-
569, DOI:10.1007/s40799-019-00307-1.
[4] Fountas, N.A., Vaxevanidis, N., V., Stergiou, C.I., Benhadj-Djilali, 
R. (2019). Globally optimal tool paths for sculptured surfaces 
with emphasis to machining error and cutting posture 
smoothness. International Journal of Production Research, 
vol. 57, p. 5478-5498, DOI:10.1080/00207543.2018.153046
8.
[5] Kong, L.B., Ma, Y.G., Ren, M.J., Xu, M., Cheung, C.F. (2019). 
Generation and characterization of ultra-precision compound 
freeform surfaces. Science Progress, vol. 103, no. 1, 
DOI:10.1177/0036850419880112.
[6] Zhang, L., Naples, N.J., Zhou W.C., Yi, A.Y. (2019). Fabrication 
of infrared hexagonal microlens array by novel diamond 
turning method and precision glass molding. Journal of 
Micromechanics and Microengineering, vol. 29, no. 6, 
DOI:10.1088/1361-6439/ab10ff.
[7] Tian, F.J., Yin, Z.Q., Li, S.Y. (2015). Fast tool servo diamond 
turning of optical freeform surfaces for rear-view mirrors. 
International Journal of Advanced Manufacturing Technology, 
vol. 80, p. 1759-1765, DOI:10.1007/s00170-015-7152-9.
[8] Khaghani, A., Cheng, K. (2019). Investigation on multi-body 
dynamics based approach to the tool path generation for 
ultraprecision machining of freeform surfaces. Proceedings 
of the Institution of Mechanical Engineers Part B-Journal 
of Engineering Manufacture, vol. 234, no. 3, p. 571-583, 
DOI:10.1177/0954405419863961.
[9] Li, D., Qiao Z., Walton, K., Liu, Y., Xue, J.D., Wang, B., Jinag, X. 
(2018). Theoretical and experimental investigation of surface 
topography generation in slow tool servo ultra- precision 
machining of freeform surfaces. Materials, vol. 11, no. 12, 
DOI:10.3390/ma11122566.
[10] Chen, X., Kang, M., Wang, X.S., Hassan, M., Yang, J. (2017). 
Tool path optimal design for slow tool servo turning of complex 
optical surface. Proceedings of the Institution of Mechanical 
Engineers Part B: Journal of Engineering Manufacture, vol. 
231, no. 5, p. 825-837, DOI:10.1177/0954405416654192.
[11] Wei, Y., Zhai, P., Chen, X.Y., He, L. (2020). Study on design and 
diamond turning of optical freeformsurface for progressive 
addition lenses. Mathematical Problems in Engineering, vol. 
2020, art. ID 2850606, DOI:10.1155/2020/2850606.
[12] Ji, S.J., Li, J.F., Zhao, J., Feng, M., Sun, C.R, Dai, H.D. (2018). 
Ultra-precision machining of a compound sinusoidal grid 
surface based on slow tool servo. Materials, vol. 11, no. 6, art. 
ID 1001, DOI:10.3390/ma11061001.
[13] Fang, F.Z., Zhang, X.D., Hu, X.T. (2008). Cylindrical coordinate 
machining of optical freeform surfaces. Optics Express, vol. 
16, no. 10, p. 7323-7329, DOI:10.1364/OE.16.007323.
[14] Lin, R.S., Koren, Y. (1996). efficient tool-path planning for 
machining freeform surfaces. Journal of Engineering for 
Industry, vol. 118, no. 1, p. 20-28, DOI:10.1115/1.2803642.