## Abstract

Freeform surface is promising to be the next generation optics, however it needs high form accuracy for excellent performance. The closed-loop of fabrication-measurement-compensation is necessary for the improvement of the form accuracy. It is difficult to do an off-machine measurement during the freeform machining because the remounting inaccuracy can result in significant form deviations. On the other side, on-machine measurement may hides the systematic errors of the machine because the measuring device is placed in situ on the machine. This study proposes a new compensation strategy based on the combination of on-machine and off-machine measurement. The freeform surface is measured in off-machine mode with nanometric accuracy, and the on-machine probe achieves accurate relative position between the workpiece and machine after remounting. The compensation cutting path is generated according to the calculated relative position and shape errors to avoid employing extra manual adjustment or highly accurate reference-feature fixture. Experimental results verified the effectiveness of the proposed method.

© 2015 Optical Society of America

## 1. Introduction

Optical freeform surface has more changes in shape than conventional components, it can be defined as any complex surface fulfilling the optical functions [1, 2 ]. Freeform optics can greatly improve the optical performance as well as reduce the system size, which can give designers more flexibility and space for innovation. For example, freeform optics can extremely fold and shorten the light path. It can produce an ultra-short throw projection with the throw distance of about 30cm, which is normally one or two meters [3]. Progressive lens can realize the multi-functions, which need to be realized by several lenses for different vision distances [4]. When the lenses are arranged in the array, the length of cell-phone lens can be significantly reduced to several hundred micrometers, which is commonly about 3mm [5].

Freeform optics needs high accuracy, such as the nanometric surface finish and sub-micrometer form accuracy, which needs the guaranty of ultra-precision machining and measurement. Commercial measuring devices provide high precision in off-machine mode. It is relatively easy to remount the conventional rotationally symmetrical surface after off-machine measurement [6]. However, it is extremely difficult for the freeform workpiece due to its asymmetric feature. Therefore, the extra reference structures or fixtures are always designed on the workpiece to provide the basis of the remounting adjustment [7, 8 ]. Figure 1 shows a typical freeform surface with very large shape deviation when there are quite small angle errors introduced in remounting. This example demonstrates the reference structures or fixtures should have relatively high accuracy to keep the adjustment precise of remounting. On the other side, on-machine measurement can be employed to avoid the errors caused by moving and remounting the workpiece [9, 10 ]. Both contact and non-contact measurement methods have been studied [11, 12 ]. The machining systems always depend on the motion axes of fabrication device to realize the profile scanning. However, the systematic errors of the machine may exist because the measuring device is placed in situ on the machine. Although they are significantly small, it is still considerable for ultra-precision machining of freeform surfaces [13, 14 ].

The off-machine measurement mode is a good choice by contrast, if the remounting problem can be avoided or reduced. There are several factors to support this initiation. For instance, the machine space is possibly limited to mount the measuring device. Especially, more parameters of freeform optics, such as the wavefront, MTF and other optical parameters need to be measured [15, 16 ]. These parameters can still only be measured in off-machine mode commonly. In addition, the freeform surface with large sag should be roughly fabricated using a precision machine first to improve the machining efficiency. After rough cutting, the workpiece should be remounted on the ultra-precision machine in order to control the un-necessary removal, so the high precision mounting is necessary. Some mechanism studies also show a need of the strict control of depth of cut (DOC) so that the high precision mounting is expressively necessary. For example, it is significant for the nanometric machining of ion implanted materials (NiIM) research due to the small implanted depth generally [17]. Therefore, the remounting technique is crucially important for machining and other fields of optical freeform surface.

An easy-operating method based on on-machine probe is proposed in this paper to obtain accurate relative position between the workpiece and machine after remounting. The compensation strategy is implemented according to the off-machine measuring and workpiece positioning results. In comparison to the current remounting methods, the proposed method avoids employing extra manual adjustment and the use of the extra highly accurate reference structures or fixtures, and can provide the high-accuracy position to improve the efficiency of the compensation process. The kernel techniques of the new compensation strategy for machining optical freeform surface are discussed as follows.

## 2 Concept and theory of new approach

As above described, the basic concept is that the compensation is based on the commercial off-machine measurement and the workpiece should be remounted in high precision. The remounting method is the use of the kernel technique in this study.

The standard model of the machined surface is the criterion for the proposed remounting method. The coordinates of the reference model are consistent or have given relative deviation with the machine coordinate generally. Therefore, the relative position between the remounted workpiece and machine can be obtained according to the achieved relative position between the remounted workpiece and the reference model. Here, the relative position is expressed as a six degrees of freedom model, which is composed of rotation matrix **R** (or denoted by three rotations, roll *α*, pitch *β* and yaw *γ*) and translation vector **T** (or denoted by three translations, Δ*x*, Δ*y* and Δ*z*).

The relative position between the remounted workpiece and the reference model can be acquired with the help of on-machine measurement. The three-axis ultra-precision turning was taken for example, which is a commonly used machine tool configuration [18]. In Fig. 2
, **T _{1}** is the translation deviation between the on-machine measurement coordinates and the machine coordinates, which can be defined through the reference ball calibration [19]. The measured points {

**} of the remounted workpiece are captured by the on-machine measurement, which are used to match the reference model in order to obtain the relative position between the remounted workpiece and the standard model denoted by**

*p*_{i}**R**and

_{2}**T**. The workpiece can be adjusted to align with the machine coordinates according to the relative position obtained. For the three-axis ultra-precision turning, the reference zero of C axis can be set at the

_{2}*γ*position. In addition, some coordinates such as

*α*and

*β*should also be set correspondingly. Because the centerline of workpiece is aligned to the spindle axis before machining,

**T**is relatively small generally. Figure 3 shows the alignment errors measured by a probe gauging along the cylinder side contour in one adjustment process, which proves that the deviations in X and Y axis are within several hundred nanometers.

_{2}In order to avoid the extra adjustment, which increases the difficulty of remounting process, the coordinate transformation machining method [20] is used to generate the compensated cutting path after mounted. The new model is established from the standard model converted by the obtained relative position. The compensated cutting path is generated based on this new model and the shape errors measured by the off-machine measurement.

Figure 4 concludes the integrated process flow, which mainly contains three process lines. The main process is a loop chain (indicated in red line), including fabrication, demounting, off-machine measurement, remounting and compensation. The other process lines support the compensation step, including the shape error from off-machine measurement (shown in yellow line) and the new model from on-machine measurement (shown in blue line). The proposed method avoids making the manual operation and achieves the accurate positioning easily at the same time.

## 3 Constrained optimization method for remounting

#### 3.1. Calculation approach

The matching relation between the measured points and the reference model is essential in the proposed method. The calculation of the matching relation is an optimization process, in which the measured points {** p_{i}**} are transformed continuously as a rigid body with six degrees of freedom, composed of rotation matrix

**R**and translation vector

**T**, until the best matching relation is found, as shown in Fig. 5 .

The optimization objective function is expressed as the minimum sum of distances between the adjusted measured points {** p'_{i}**} and the corresponding model points {

**},**

*q*_{i}The optimal transformation can be calculated by the nonlinear least squares algorithm, such as Gauss–Newton or Levenberg–Marquardt algorithm [21]. However, it is difficult to find the globally optimal solution because of the large solution space. The proper constraint conditions should be set to avoid falling into the local optimal solutions. Two kinds of constraint conditions can be put in the optimization process. One kind of conditions ensures the non-change shape in the transformation process, which is controlled by the orthogonality of rotation matrix **R**,

*r*(

_{i}*i*= 1~9) are the items of matrix

**R**. Another kind of constraint condition guarantees that the transformation space of the measured points should conform to the real measurement status, which is related to the machining and measurement methods. For example,

**T**is relatively small generally for the three-axis ultra-precision turning. The constraints can be added as the following,where

*t*(

_{i}*i*= 1~3) are the items of vector

**T**.

#### 3.2. Simulation analysis and evaluation

The simulation is implemented to analyze the proposed remounting method in this section for the ultra-precision three-axis turning. The measured points are simulated by the six-degrees of freedom transformation of the typical surface. The relative positions calculated by the proposed method are compared with the given transformation parameters to prove the calculation accuracy. The given transformation parameters are shown in Table 1 . These parameters properly accord with the features of machining configuration, where the yaw is a random number in the range of [-180°, 180°] and other values are relatively small.

### 3.2.1. Simulation results for different types of surfaces

Three types of surfaces, including plane surface, continuous smooth surface and arrayed-structure surface, are analyzed in the aperture of 40 mm as shown in Table 2 . In the table, the errors are the peak to valley (P-V) values of the deviations between the transformed measured points and the corresponding model points.

The results shown in Table 2 illustrate that all of three surfaces can achieve good remounting accuracy. The number of the simulated measured points is 10000. The plane surface has the best accuracy, but the calculated relative position does not agree with the simulated transformation because there is no constraint in the surface shape. Other two kinds of freeform surfaces can get good remounting accuracies of less than 100 nm in P-V value, and their relative positions are consistent. The consistencies of the parameters are sensitive to the surface shape, which should be considered greatly in the remounting process. The yaw *γ* is the most sensitive parameter for the turning method. In addition, the number of sensitive parameters increases when the surface curvature varies significantly. The arrayed-structure surface is an example to support the conclusion.

### 3.2.2. Simulation results for different number of measured points

This simulation also aims to provide the required number of sampled points. Table 3 shows the simulation results for two kinds of freeform surfaces mentioned in the above section. The results prove that the number of measured points has little effect on the accuracy of the proposed method. The less number is better for high efficiency of measurement under the condition of minimum number satisfying the optimization calculation.

As above analyzed, only some scattered measuring points are required in the optimization process. Therefore, any on-machine measurement approach [9, 10 ] is suitable for the proposed method. The scanning probe method, such as a contact ruby probe or non-contact laser point, is a good choice considering time and cost consumption.

### 3.2.3. Simulation results for the measured points with errors

The errors are unavoidable in this process, mainly including measurement errors and form errors of the workpiece. In this simulation, the effectiveness of the proposed method is verified with the induced errors. Table 4 shows that the errors between the adjusted measured points and the corresponding model points agree well with the errors induced. The results prove that the proposed method is effective and accurate for the ultra-precision machining of freeform surfaces.

## 4. Experiments and results

The following experiments were performed to verify the proposed method. The accuracy of the kernel optimization algorithm and remounting the position needs to be evaluated in comparison. Finally one integrated compensation experiment was implemented to verify the effectiveness of this compensation strategy.

#### 4.1. Accuracy test of optimization method

A freeform surface, expressed by XY polynomials, was fabricated by the ultra-precision three-axis turning. A commercial instrument, Panasonic UA3P-5, was used to measure it in the range of 80mm × 80mm. The form error was evaluated by the analysis software provided by the commercial instrument, as shown in Fig. 6(a) . The evaluated form error of P-V was 2.485μm. The measured points were also evaluated by the proposed optimization method introduced in Section 3. The evaluated form error of P-V was 2.264μm, as shown in Fig. 6(b) and Fig. 6(c). These two evaluated results are in good accordance in the aspects of the error map and value, which proves that the proposed optimization method has wide availability and high accuracy.

#### 4.2. Remounting position accuracy evaluation

A saddle surface *z* = (*x*/51.3736)^{2} - (*y*/51.3736)^{2} was tested in the following two subsection. The surface depth map with the aperture of 40mm is illustrated in Fig. 7(a)
. The workpiece was fabricated by one ultra-precision five-axis turning machine in slow slide servo mode [18]. Then the workpiece was detached from the machine and then measured by the Form Talysurf profilometer (AMETEK Inc.). It was then remounted freely. The on-machine profilometer with air bearing was used to measure the machined surface for obtaining the remounting position. A Moore Nanotech’s on-machine measuring probe was adopted as shown in Fig. 8
. Its probe with Linear Variable Differential Transformer (LVDT) sensor has been verified to have good linearity and a resolution of 20nm in the measurement range of ± 300μm. The reference step measurement has proved that the uncertainty and standard deviation are about 9‰ and 10nm respectively. The 1μm step measurement results are shown in Fig. 9
.

The form error was measured by the off-machine profiler and evaluated by the proposed optimization method. The P-V value is measured as 4.047μm, whose error map is shown in Fig. 7(b). The relative position after remounting was measured and calculated in on-machine mode as the above described. The uniform sampling was used for the scanning strategy, and there are 49 sampling points captured. The rotation positions are optimized as listed in Table 5 , which indicates the large yaw, and the rather small roll and pitch value. The translation position are also calculated as shown in Table 5, which demonstrates there is a small offset in Z axis between the measurement system and fabrication machine. To verify the calculation accuracy of relative positions, one more measurement was implemented after remounting. There are 309 uniform sampling points, which are converted according to the calculation relative position. The converted points are compared with the reference model directly. The comparison result is shown in Fig. 10(a) . The points conform to the reference model well. The deviation is calculated as 4.724μm, as shown in Fig. 10(b). The deviations are in the same level with the off-machine results, which are in good accordance with the form error.

#### 4.3. Compensation machining

To verify the proposed compensation strategy, the remounted workpiece was fabricated to compensate the form error. The compensation path is designed according to the coordinate transformation machining method, in which the calculated relative position is considered as well as the compensated errors. Therefore, there is no need to adjust the workpiece position consistent with the machine by extra manual operation. The compensation path and the source cutting path are displayed in Fig. 11(a) , which demonstrates the difference of two paths. The compensation process includes two cutting steps. The cutting depth of rough cutting is 5μm, which aims to remove the shape deviation due to the remounting. And the precision cutting adopts the cutting depth of 2μm to ensure the cutting accuracy. Finally, the form error after compensation was measured by the off-machine measurement with the P-V value of 1.482μm, as shown in Fig. 11(b). The experimental result proves that the novel compensation strategy is effective to compensate the shape error.

## 5. Conclusions

A novel compensation method with a combination of on-machine and off-machine measurement is proposed for machining optical freeform surface. The on-machine measurement is proposed to realize the high accuracy and easy-operating remounting method for freeform surface. The constrained optimization method provides the basis for finding the relative position between the remounted workpiece and the reference model. The simulated and experimental results verify the accuracy of the proposed method, which highlights the following advantages.

- (1) The proposed method realizes the remounting of freeform surfaces, which avoids the complicated reference-featured fixture and manual operation. The coordinate transformation machining method is utilized in the compensation path generation to avoid the extra manual adjustment.
- (2) The proposed method is used in the compensation machining by off-machine measurement. It provides the capability of remounting workpiece in high accuracy, which can meet the requirement of other parameters in off-machine measurement.
- (3) The proposed method has great potential for other off-machine applications of freeform, such as the rough prefabrication and the specimen positioning for the machining mechanism research in high accuracy.

## Acknowledgments

The authors express their sincere thanks to Y.B. Lu and Y. X. Xiang for the enlightening discussion and invaluable advices on the experiments. This work has been funded by the State Key Development Program of Basic Research of China (“973” Project Grant No. 2011CB706700), the National Natural Science Foundation of China (Grant No. 51375337) and the Special-funded Program on National Key Scientific Instruments and Equipment Development of China (Grant No. 2012YQ06016501).

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