# A Differential Measurement System for Surface Topography Based on a Modular Design
## 2 Overall System Design  
The system configuration of the proposed design is shown in Figure 2. The overall dimension of the system is L 890 mm \(\times\) W 820 mm \(\times\) H 675 mm. The measurement range is 100 mm \(\times\) 100 mm \(\times\) 50 mm. Like the first prototype, this system was built of aluminum alloy. The central stage was supported by two sets of 2D sliders and driven by a pair of actuators (Zolix uKSA 100, Zolix Instruments CO.,LTD, Beijing, China). Each uKSA 100 is able to achieve a travel range of 100 mm, which determined the measurement range of the presented system.  
Figure 1: Graphic abstract.  
A confocal sensor, Precitec CHRocodile SE (Precitec Group, Germany), was mounted on the main spindle to measure the height at each location of the test piece. This confocal sensor was able to achieve 3 nm resolution in a measurement range of 600 \(\upmu\)m. An optical flat [32] (\(\varphi\) 300 mm \(\times\) T 30 mm, K9 glass) was placed in the center of the stage as the geometric datum. A film interferometer was developed in order to configure a differential measurement system with the confocal sensor. During the lateral movement, the film interferometer was used to compensate for the straightness error by measuring the fluctuation of the optical plat from the bottom.  
The XY positions were measured by linear encoders, which were coupled with the central stage through contact kinematic pairs. Each contact kinematic pair consisted of two optical flats, which helped to minimize the coupling errors. The linear encoders only moved along the measurement axes. In the XY plane, therefore, a static coordinate system was formed by the linear encoders, and the system complied with the Abbe principle [23; 24].  
In this system, the position measurement accuracy relied on the datum features formed by the optical flats, instead of the mechanical translation components. All of the linear guides used in this system were within the ordinary accuracy range.  
The geometric error factors for the height measurement are shown in Figure 3. The actual height of the measurement point could be expressed as:  
\[h=h_{1}-h_{2}-h_{3}-\left(\frac{T}{cos\theta}-T\right). \tag{1}\]  
Figure 3: _Cont._  
Figure 2: System design: (**a**) Side view, (**b**) Top view.  
In Equation (1) \(h_{1}\) is the height measured by the confocal sensor, \(h_{2}\) represents the motion error caused by the linear guides, and \(h_{3}\) is the height fluctuation due to the flatness error of the optical flat. T is the thickness of the optical flat and \(\theta\) is its tilting angle during the movement.  
In this system, the flatness of the optical flat was \(\pm 25\) nm. The thickness of the optical flat was 30 mm. By assuming that the angular motion error was no greater than 20\", the maximum height measurement error \(T(1-cos\theta)\) could be calculated to be 0.14 nm. Therefore, the values of \(h_{3}\) and \(T(1-cos\theta)\) were very insignificant and Equation (1) could be simplified as:  
\[h=h_{1}-h_{2}, \tag{2}\]  
where \(h_{2}\) was mainly contributed by the straightness error of the linear guide and it could be measured by the film interferometer.  
Instead of an all-in-one structure, the concept of modular design was employed in the proposed system. As shown in Figure 2, the system was assembled with a few functional modules:  
1. An actuating module, which consisted of two motorized stages.
2. Supporting modules, each of which consisted of pre-assembled 2D sliders
3. A differential height measurement module, which consisted of a confocal sensor and a film interferometer
4. A displacement measurement module, which consisted of an optical gratings couple with the central stage through optical flats.
5. A datum module, which consisted of three optical flats.  
Since these functional modules were all commercially available components, no special design or assembly processes were needed in the proposed system.