Matching Measurement Strategy and Form Error Compensation for Freeform Surface Machining Based on STS Turning

Abstract

The freeform surface quality is limited by the measurement, and form error cannot be convergent via compensation machining. This paper proposed a surface matching measurement strategy based on the least squares principle and iterative precision adjustment to precisely obtain surface form error after manufacturing by single point diamond turning. Through the coordinate transformation of translation and rotation, the measured surface was aligned with theoretical surface at the same coordinate system. The corresponding simulation was carried out to verify the performance of the proposed method, and the simulation results indicated that this method can achieve accurate alignment in a sub-nanometer range. Finally, with XY polynomial freeform surface as the ideal surface, compensation experiments were undertaken. The form error of freeform converged continuously after compensation machining for three times, during which the form accuracy of PV and RMS were down to 335 nm and 34 nm respectively from 1.4 um and 173 nm. The results showed that capability of the proposed compensation method was verified and the form accuracy could be improved effectively.

1. Introduction

To improve the performance of optical system and reduce the system size, the requirement of surface shape of optical element becomes more and more fastidious, such as the one of freeform surface. Optical freeform surfaces are widely used in head-up displays, lighting optics, aerospace and biomedical imaging [1,2,3]. However, a high accuracy of freeform surface fabrication is still a challenge for ultra-precision freeform surface measurement and machining [4,5].

For the machining of freeform optical surface, the slow tool servo (STS) turning based on single point diamond turning is a choice of effectiveness and efficiency [6]. In slow tool servo turning, the motion of the tool on the Z axis synchronizes with the radial motion of the X axis and the rotation of the machine tool on the C axis. Therefore, the machining of non-rotationally symmetric surfaces can be easily achieved, and in this way can be comparatively efficient [7,8]. Many scholars have carried out corresponding researches. Fang et al. [9] made a review on freeform surface manufacturing and studied the capability of slow tool servo. Vinod Mishra et al. [10] developed a tool path compensation to reduce the form error of freeform surface. We can see that there have been many other researches on how to improve the surface quality of freeform optics fabricated by STS [11,12], and pertinent studies have made this field become a research hotspot.

One crucial key to extending the edge of the field is that it is necessary to obtain precise measurement of freeform surface first as feedback for the next compensation machining [13,14]. Due to the geometric complexity of freeform surface, the characterization of surface quality lacks definitive standards and techniques. There are many types of ultra-sophisticated measuring equipment, and notwithstanding freeform surface shape can be measured, the shape error as the compensation feedback cannot be attained directly [15]. Effective compensation feedback can be obtained only via precise re-mounting of freeform surface. Initially, confirm that the consistency of measurement coordinates with design coordinates. Then the projection of the measured point on the theoretical coordinates after coordinate transfer is calculated. Ultimately by computing the distance of measured surface from the theoretical design surface in theoretical coordinates, the form error as compensation feedback is obtained.

Related studies have explored the matching method of the freeform surface. Cheung et al. [16] presented an integrated form characterization method based on the five-point pre-fixture, dual cubic B-spline and an iterative precision adjustment algorithm with coordinate transfer to measure the form accuracy of ultra-precision freeform surface. Wang et al. [17] proposed a fiducial-aided on-machine positioning method to precisely position optical freeform surfaces during the precision manufacturing cycle, by using the developed on-machine measuring devices to get the accurate positions of the fiducials after remounting in the machining coordinate system. Tong et al. [18] developed a closed-loop FTS system to improve freeform machining accuracy, surface sampling and reconstruction strategies and robust surface filtration algorithms that are adapted to regulate the data flow for both freeform surface characterisation and the optimisation of compensation toolpath from machined error maps. Zhang et al. [19] proposed a new compensation strategy based on the combination of on-machine and off-machine measurement. These methods intend to realize optimal alignment by moving and rotating coordinates so that the feature points, lines, and surfaces of the measured surface and the designed surface overlap completely. Because the accuracy of these method for characterizing freeform surfaces with form error is not every high, it is difficult for the final surface machined by compensation to reach sub- micron precision [20,21].

In this paper, a new method based on the least squares principle and iterative precision adjustment is put forward to obtain surface form error, in the meantime, the compensation method of STS freeform processing was developed to improve the surface quality. By computer simulation, the accuracy and feasibility of the matching method is verified. Also, the freeform surface compensation machining experiment results based on STS suggest that the method is effective for improving surface form accuracy.

2. Modeling and Experimental Procedures

This compensation processing is based on high-precision surface error detection results. We modify the processing program according to compensation algorithm, which was then carried out repeatedly until the form error is within the tolerance. Matlab software was used to simulate and modify the processing program. The challenge here is to obtain form error from measured data of freeform surface for the measured and theoretical surface represented in different coordinate systems. Coordinate offset between the two systems needs to be calculated and the extent of difficulty depends on the degree of freedom of surface, thus it is challenging to match the measured and ideal surface for freeform optics.

As described above, ultra-precision compensation machining is based on high accuracy alignment of the measured and designed surface. The aim of alignment is to make the measured surface as close to the theoretical surface as possible. To achieve the best alignment, a rough adjustment based on tooling and precise alignment by iterative adjustment is adopted.

The detection surface and the theoretical surface are represented in different coordinate systems. There are spatial position deviations in six degrees of freedom, as shown in the Figure 1. Surface matching is used to make the measuring coordinate system coincide with the theoretical coordinate system. In the actual matching process, it is more suitable to make the measured surface approximates the theoretical to the maximum extent as far as possible.

The first step is to apply the fixture for rough positioning. As showed in Figure 2, there is a plane as the mark which is parallel to the plane comprised by axis X, axis Z, and measuring and theoretical coordinate systems are all based on the mark. Thus, the theoretical coordinate system and measuring coordinate system are quite alike.

The process of fine matching requires translation and rotation operation of six degrees of freedom, to achieve accurate alignment between the surfaces of the measured and theoretical systems. In the spatial coordinate system, the rotation and translation coordinate transformation corresponds to the rigid body transform. According to the transfer matrix, the point P(x, y, z) in the measured surface changed to point P′ (x′, y′, z′) in the theoretical coordinate system, as shown in the Equation (1).

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cc}{R}_{3\times 3}& \begin{array}{c}tx\\ ty\\ tz\end{array}\\ \begin{array}{ccc}0& 0& 0\end{array}& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right]$$

(1)

where tx, ty and tz represent the shifts in the x, y and z directions respectively. R_3×3 is the rotation matrix, which is defined as Equation (2):

$$R_{3\times 3}=\left[\begin{array}{ccc}\cos \gamma & -\sin \gamma & 0\\ \sin \gamma & \cos \gamma & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \beta & 0& \sin \beta \\ 0& 1& 0\\ -\sin \beta & 0& \cos \beta \end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \alpha & -\sin \alpha \\ 0& \sin \alpha & \cos \alpha \end{array}\right]$$

(2)

where $\alpha$, $\beta$ and $\gamma$ are the rotation angles along the x, y and z axes respectively. The point ${\mathrm{Q}}_{ij}\left(X_{ij},Y_{ij},Z_{ij}\right)$ is the closest point of the theoretical surface to the point $P_{ij}\left({X^{\prime }}_{ij},{Y^{\prime }}_{ij},{Z^{\prime }}_{ij}\right)$, the deviation can be calculated by Equation (3).

$$F\left(\alpha ,\beta ,\gamma ,tx,ty,tz\right)={\displaystyle \sum }{\left|Q_{ij}-P_{ij}\right|}^2.$$

(3)

The best matching condition is defined as the minimum value of the function, namely, the sum of the distance between each point on the measured surface and the corresponding projection point on the theoretical surface are all in their minimum. Through multi-parameter optimization and based on the least squares principle and iterative precision adjustment, the optimal position parameters can be obtained and then the surface matching is complete. In such cases, the surface form error function is

$$E=Z_{ij}-{Z^{\prime }}_{ij}.$$

(4)

The evaluation parameters PV and RMS of surface form error were defined as Equations (5) and (6):

$$PV=max\left(E\right)-min\left(E\right)$$

(5)

$$RMS=\sqrt{\frac{1}{m}{\displaystyle \sum }_{i=1}^m{E}_i}$$

(6)

where m is the number of points on the measured surface.

During the process of freeform surface compensation machining, it is necessarily required to judge whether the surface form error meets the requirement. When the form error exceeds the tolerance, the obtained surface form error will be used to compensate the tool cutting path. The flow chart of the compensation process is shown in Figure 3. It should be noted that form error is regarded as the surface to be removed next in the process of surface matching. The surface error is reversed and superimposed on the initial ideal surface which will act as the new freeform surface to be processed. Because of the existence of random error, it is estimated that there will be repeated fluctuations in error, and the surface form error needs to be compensated several times until it is within the tolerance.

Simulation Study

To evaluate the effectiveness and accuracy of the matching algorithm, theoretical simulation is carried out. An ideal freeform surface is designed as the theoretical surface, and the measured surface was obtained by adjusting the designed surface through six parameters of the coordinate transformation matrix. Then the accuracy of the simulation algorithm was calculated. The simulation flow chart is shown in Figure 4.

To begin with, transfer the ideal surface in theoretical coordinate system to the measurement surface with coordinate transformation T₁. Then match the ideal and measured surface by the proposed algorithm. After surface matching, subtract the theory shape from the measurement in the same coordinate system to get the form error ε, and the coordinate transformation T₂ is calculated. As the measured surface is obtained from the theoretical surface through coordinate transformation, the ideal form error ε is zero, and the coordinate transfer matrix T₁ and T₂ should be the same. Therefore, the value of form error ε and the deviation between T₁ and T₂ could verify the accuracy of theoretical model and the precision of matching algorithm.

The designed ideal freeform surface is XY polynomial, the surface equation is shown in Equation (7), and the surface parameters are shown in

Table 1. Surface shape parameters.

Parameters	Value	Parameters	Value
Diameter	10 mm	y2 (c6)	0.0002
c	0.01	x3 (c7)	0.00112
k	0	x2 × y (c8)	0
x (c2)	−0.03	x × y2 (c9)	−0.0004
y (c3)	0.007	y3 (c10)	0.0002
x2 (c4)	−0.0001	x4 (c11)	0
x × y (c5)	−0.005	x3 × y (c12)	0.0005

. The parameters of coordinate transfer matrix T (α, β, γ, tx, ty, tz) are varied to obtain the corresponding simulation results. They are divided into 10 groups; the first four groups only perform translation coordinate transformations, the second four groups only perform rotation coordinate transformations, and the last two groups include translation and rotation. The results of simulation are shown in

Table 2. Simulation results of form error with different coordinate transfer matrix $T_1$.

Parameters	$\mathit{t}\mathit{x}/\mathit{m}\mathit{m}$	$\mathit{t}\mathit{y}/\mathit{m}\mathit{m}$	$\mathit{t}\mathit{z}/\mathit{m}\mathit{m}$	$\mathit{\alpha }/\mathit{r}\mathit{a}\mathit{d}$	$\mathit{\beta }/\mathit{r}\mathit{a}\mathit{d}$	$\mathit{\gamma }/\mathit{r}\mathit{a}\mathit{d}$	$\mathit{P}\mathit{V}/\mathit{n}\mathit{m}$	$\mathit{R}\mathit{M}\mathit{S}/\mathit{n}\mathit{m}$
Translation	1	0	0	0	0	0	0.281	0.068
	0	1	0	0	0	0	0.136	0.033
	0	0	1	0	0	0	0.148	0.033
	0.2	0.3	0.5	0	0	0	0.226	0.058
Rotation	0	0	0	0.5	0	0	0.119	0.029
	0	0	0	0	0.5	0	0.074	0.019
	0	0	0	0	0	0.5	0.165	0.039
	0	0	0	0.3	0.4	0.5	0.057	0.014
Translation and rotation	0.5	1	0.8	0.5	0.5	0.6	0.105	0.027
	1	1	1	0.5	0.5	0.5	0.120	0.030

, and the form errors are shown in Figure 5.

As shown in Figure 5, when the surface is transformed by translation or rotation, the matching error is tiny. However, the matching error is different with different coordinate transfer matrix T1, which represents that different coordinate transformations have different effects on the results of matching model. The simulation error of coordinate rotation is smaller than that of coordinate translation, although they are on the same order of magnitude. Minimum error occurs when there are three directions of coordinate rotation. Although coordinate transformation involves translation and rotation of six degrees of freedom, the matching error is tiny, which verifies the high precision of the matching model.

$$z=\frac{c\left(x^2+y^2\right)}{1+\sqrt{1-\left(1+k\right)c^2\left(x^2+y^2\right)}}+{\displaystyle \sum _{i=0}^p{\displaystyle \sum _{j=0}^p{C}_{i,j}{x}^i{y}^j}}$$

(7)

It can be seen from Table 2 and Figure 5 that the freeform surface matching algorithm possesses sub-nanometer accuracy which represents the error from the theoretical model. Compared with the submicron precision of ultraprecision machining, this error is negligible. Thus, the validity of theoretical model is verified. The form error of PV is around 0.2 nm, and the value of RMS is about 0.06 nm. The error varies with six spatial parameters, and rotational transformation error is slightly less than translational transformation error. Moreover,

Table 3. Coordinate transfer matrix $T_2$ calculated through surface matching.

Parameters	$\mathit{t}\mathit{x}/\mathit{m}\mathit{m}$	$\mathit{t}\mathit{y}/\mathit{m}\mathit{m}$	$\mathit{t}\mathit{z}/\mathit{m}\mathit{m}$	$\mathit{\alpha }/\mathit{r}\mathit{a}\mathit{d}$	$\mathit{\beta }/\mathit{r}\mathit{a}\mathit{d}$	$\mathit{\gamma }/\mathit{r}\mathit{a}\mathit{d}$
Translation	1	4.94 × 10−9	−4.32 × 10−10	−5.51 × 10−9	3.99 × 10−10	−1.55 × 10−9
	1.94 × 10−9	1	3.70 × 10−9	2.08 × 10−10	−2.62 × 10−9	7.46 × 10−10
	−6.02 × 10−10	3.05 × 10−10	1	−8.95 × 10−10	−2.43 × 10−9	1.23 × 10−9
	0.2	0.3	0.5	−3.84 × 10−9	−2.41 × 10−9	4.03 × 10−11
Rotation	−2.74 × 10−9	−8.13 × 10−10	2.04 × 10−9	0.5	1.78 × 10−9	1.02 × 10−9
	2.73 × 10−11	−2.28 × 10−9	−3.76 × 10−9	−4.97 × 10−10	0.5	3.29 × 10−10
	2.24 × 10−9	5.68 × 10−9	1.58 × 10−9	−5.53 × 10−11	2.60 × 10−9	0.5
	−3.67 × 10−9	−1.60 × 10−9	4.59 × 10−9	0.3	0.4	0.5
Translation and rotation	0.5	1	0.8	0.5	0.5	0.6
	1	1	1	0.5	0.5	0.5

shows the coordinate transfer matrix T₂ that calculated through surface matching. As discussed above, the deviation between matrix T₁ and T₂ is almost close to zero. This error is generated by the calculation algorithm which represents the system error of the proposed method. As the simulation results show, the error fluctuates in sub-nanometer range along with the change of the parameters. Compared with the precision of turning, the error generated by the matching algorithm is negligible, which infers that the matching method is adequately sufficient to get form error of freeform surface, and the accuracy of the method is verified in the sub-nanometer range.

3. Experimental Study

The compensation experiment was carried out to verify the effectiveness of the proposed method, where an optical freeform surface was employed to test the feasibility of the compensation processing. The designed surface is given as Equation (7). The surface was machined by STS based on Precitech Nanoform 700 ultra single point diamond turning machine, and was measured by the newview8200 measurement system from Taylor Hobson Co., Ltd., in UK. The workpiece material is Al, and the tool parameters and machining parameters are shown in the

Table 4. Cutting tool parameters and machining parameters.

Cutting Tool Parameters			Machining Parameters
Radius/mm	Rake/°	Clearance angle/°	Rotate speed/rpm	Feedrate/mm/r	Cut depth/um
1	0	20	40	0.005	2

.

The freeform surface was measured after the first processing. Figure 6 shows the machined surface and its form error after the matching process. As shown in Figure 6b, the form error between the machined surface and the designed surface is relatively small, and the PV and RMS value were 1.4 um and 173 nm respectively. Such precision could not meet the requirements of ultra-precision fabrication, the compensation machining is a necessary process to enhance the form accuracy of the freeform surface. The compensation machining process is as follows. After the surface is measured, take the measurement result as the surface after coordinate transformation of ideal freeform surface. Then the measured surface is matched with the ideal surface by matching algorithm. The form error is derived by the algorithm and taken as the surface to be removed in the next step. The Z axis data of the surface shape error is inverted, and then added to the Z axis data of the ideal freeform surface. It is used as the surface shape to be processed in compensation machining. Then reprogram to generate a new processing program and start compensation machining. The compensation process is designed according to the form error calculated, and the cutting parameters are the same as the first processing, compensation result is shown in Figure 7a. The form error was reduced to PV value of 560 nm and RMS value of 55 nm. It is obvious that the surface quality was improved effectively.

In addition, there are some factors affecting the performance of compensation processing, such as random errors in the machining process. Only a one-time compensation failed to achieve the desired result, so the second compensation processing was carried out. As shown in Figure 7b, in the second compensation, PV value is reduced to 352 nm, and RMS value is reduced to 23 nm, the surface quality is further improved. That is the same as the process of polishing. It can be said that the form error compensation is a process of continuous convergence. As a result, it is necessary to carry out compensation machining cyclically until the form error meets the requirement.

The third compensation processing was conducted, and the result is shown in Figure 8. The PV value is 335 nm and the RMS value is 34 nm. This time the PV didn’t change much, but the RMS increased slightly. As described above, the form error is fluctuant in the process of compensation so this phenomenon is normal in the error convergence process, for the random error such as vibration of the machine or tool wear is likely to affect performance in the submicron accuracy range. In a general perspective, the presented matching method and compensation strategy is effectively to improve the surface quality of freeform surface in high precision.

4. Conclusions

Freeform surface is non-rotational symmetric, and it is comparatively hard to process it in high precision. The major challenge is from the form error measurement which was described in this paper. To improve the machining accuracy of freeform optical, a surface matching measurement strategy and form error compensation method for freeform surface machining based on STS turning are proposed. The surface matching method is based on the least squares principle and iterative precision adjustment to precisely obtain surface form error. The effectiveness and accuracy of the matching algorithm have been demonstrated through computer simulations. The simulation results show that the system error caused by the algorithm is within the sub-nanometer range, and indicate that matching method can realize precise alignment between the measured and theoretical surfaces in nanometer accuracy range. With the base of precise alignment, the form error is reduced by compensation machining, and the continuous convergence of error processing verifies the capability of compensation method. The experimental results demonstrate the accuracy and effectiveness of the proposed method. This study appears greatly promising and is beneficial for ultraprecision freeform surface manufacturing.