# A Differential Measurement System for Surface Topography Based on a Modular Design
## 4 Analysis of the Degree of Freedom  
In order to allow a long travel range and to accommodate a differential measurement setup, the proposed system needed separate supporting and guiding mechanisms, as discussed in Section 2. Furthermore, the Abbe error-free design may have also introduced a separate guiding mechanism [31; 33; 34]. To avoid an over-constrained mechanism, a strict alignment and adjustment process was needed. The straightness and parallelism of the linear guides are very critical to such systems. In order to address this challenge, the proposed modular system was developed based on DoF analysis:  
As shown in Figure 8, the key points of designing and assembling the system were those listed below:  
1. There were one actuating module and two supporting modules in this system. Each supporting module consisted of two perpendicularly placed sliders, which were pre-assembled. This 2D sliding pair could be considered a plane contacting pair that allowed free movement in a horizontal plane despite possible in-plane placement errors.
2. The sliders were well leveled to the horizontal reference, assisted by an LVDT (Linear Variable Differential Transformer)
3. The stage top was rigidly connected with the actuating module, and it was coupled with the two supporting modules through ball hinges.  
Figure 7: Relationship between phase shift and distance.  
The following paragraphs provide the DoF analysis of the mechanism.  
Traditional DoF theory does not cover the scenarios of imperfect straightness or parallelism, and parallel linear guides are often considered redundant constraints [35]. This methodology, however, does not apply to precision positioning systems. Based on the traditional theory, the structure shown in Figure 8 could be simplified to the mechanism shown in Figure 9a. When imperfect straightness and parallelism were considered, the linear guides had to be represented by two-point contacting pairs instead of sliding pairs, as shown in Figure 9b. The DoF of the mechanism shown in Figure 9b could be expressed by:  
\[\text{DoF}=3\cdot\text{n}-2\cdot\text{p}-1\cdot\text{q}. \tag{16}\]  
In Equation (16), \(n\) is the number of components: \(n=6\). \(p\) is the number of revolving pairs (ball hinges M and N), each of which was constrained to 2 DoF: \(p=2\). \(q\) is the number of contacting pairs A to L, each of which was constrained to 1 DoF: \(q=12\). The DoF of this mechanism could then be calculated to be two, which met the requirement of 2D motion.  
Therefore, it was concluded that when the sliders were all well leveled and the 3D model could be projected to a 2D plane, an over-constrained mechanism would not be a concern for the proposed  
Figure 8: XY motion system.  
Figure 9: Mechanical model for DoF analysis: (**a**) Theoretical model: ideal components and ideal assembly, (**b**) Actual model: imperfect straightness parallelism.  
system. When a bigger measurement range is demanded in the future, this system can be easily reconfigured by replacing the actuating and supporting module with longer travel ranges.  
With the assistance of an LVDT, the error of leveling could be controlled within a micrometer level, which would not affect the motion smoothness. The motion error due to the imperfect leveling can be detected by the film interferometer and compensated accordingly.  
When a bigger measurement range is demanded, the system may need more supporting modules. As shown in Figure 9, one more supporting module would introduce two more components, one more revolving pair, and four more contacting pairs. By using Equation (16), it could be calculated that the additional DoF introduced by one more supporting module was zero, which meant more additional supporting modules would not affect the system DoF. With this modular design, therefore, the system had good scalability.