# Surface Texture Measurement on Complex Geometry Using Dual-Scan Positioning Strategy
## 4 Surface Tracing Strategy for Long Scan Length
### Surface Tracing Algorithm  
For high-resolution surface measurement, the confocal sensing range in the vertical direction is typically less than 1 mm. The Keyence LT9011 laser confocal sensor used in this study has a vertical sensing range of 0.6 mm. When measuring rough surfaces with significant curvature, the measurement is very likely to fail due to insufficient sensing range, as shown in Figure 4a. In order to ensure surface information was always captured within the sensing range, an adaptive surface tracing algorithm was developed.  
The methodology is illustrated in Figure 4b. The sensor moved in piece-wise mode. Each unit travel length was 1 mm. Since the oscillating unit inside the senor was able to generate a local scan length of 1.1 mm, the scanning range of the neighboring scans had an overlapping section of 0.1 mm. This overlapping section was used for profile reconstruction based on the best fitting algorithm [25].  
In this system, a vertical positioner (LS-110, Physik Instrumente GmbH & Co. KG, Karlsruhe, Germany) was used to position the sensor to ensure that at each lateral scanning location, the data capture was within the sensing range. The positioner LS-110 was able to achieve a maximum speed of 90 mm/s and a positioning resolution of 50 nm in a range of 26 mm. The actual speed needed in this study depends on the lateral movement speed and the height variation of the measured surface. Since the lateral scanning speed generated by the internal positioner was 2 mm/s, this vertical positioner LS-110 had sufficient capability for most use cases.  
At each location, the sensor LT9011 collected height information over a scan length of 1.1 mm. With the lateral position information determined by the sampling spacing and sequence number, a second-order polynomial \(f(x)\) was obtained based on least-squares fitting. Then, the height to position the sensor for the next scan was determined as  
\[h_{j+1}=\frac{1}{2}f^{\prime}(x_{n})+f(x_{n}). \tag{2}\]  
where \(f^{\prime}(x)\) is the first derivate of \(f(x)\), \(l\) is the lateral positioning increment and \(n\) is the number of data points acquired in \(l\). In this study, \(l\) and \(n\) were set as 1.1 mm and 551, respectively.  
Figure 4: _Cont._  
The vertical positioner has an embedded optical encoder to record the position. Therefore, the height \(z_{i,j}\) measured by the system included two portions: the height measured by the senor LT9011 and the height read from the optical encoder.  
Figure 5a shows the setup for measuring a free-form surface. The sample was carried by a linear stage (TSA100-B, Zolix Instruments Co., Ltd., Beijing, China), the displacement of which was measured by an in-house developed image grating system [26; 27]. Using this integrated system, the scanning range can be expanded to 100 mm. In this study, at each location, time taken for positioning and local scan was 1 s.  
Figure 5b shows the scanning result. The vertical measurement range can be expanded using the proposed surface tracing algorithm.  
Figure 4: Surface tracing strategy to extend the measurement range: (**a**) measurement out of sensing range; (**b**) surface tracing.  
Figure 5: Free-form surface measurement: (**a**) system configuration; (**b**) scan data.