## 1 Introduction
### A Technique to Evaluate Angular Misalignments of a Small Cylinder  
Regarding the results of numerical calculations described above, angular misalignments \(\theta_{X}\) and \(\theta_{Z}\) of a small cylinder need to be reduced as much as possible. It is thus necessary to establish a technique to detect \(\theta_{X}\) and \(\theta_{Z}\) in advance of the roundness measurement of a workpiece. However, regarding the small dimensions of a workpiece to be measured by the stitching linear-scan method, it is not so easy to apply conventional angle sensors such as an optical autocollimator [19] or a multi-axis laser autocollimator [20] for this purpose. To address the issue, attempts are made in this paper to utilize the linear scan surface form stylus profiler, which is employed for roundness measurement in the stitching linear-scan method, for the detection of \(\theta_{X}\) and \(\theta_{Z}\). Now we consider a positioning system shown in Figure 8a composed of a tilt stage, a \(Z\)-rotary stage, and a workpiece holder. The tilt stage, the rotary stage, and the workpiece holder are stacked and assembled as a holder unit. Figure 8b shows a photograph of the developed holder unit. The holder unit can be mounted on a linear stage. In the proposed method, \(\theta_{X}\) is detected through a two-step procedure. In the first step, the workpiece axis is aligned to be parallel with the \(X\)-axis as shown in Figure 9a. By scanning over the workpiece holder surface with the stylus, profile data (\(x_{i\_\  
In the second step, the holder unit is rotated \(90^{\circ}\) about the \(Z\)-axis on the linear stage through the demounting and remounting operations, as shown in Figure 9b. It should be noted that \(\theta_{\text{Holder}}\) does not change during the operations since the components in the holder unit are rotated together without separation. Denoting the inclination angle of the top surface of the linear stage about the \(X\)-axis with respect to the datum surface of the surface form stylus profiler as \(\theta_{\text{Linear\_X}}\), the angular position of the workpiece holder surface about the \(X\)-axis with respect to the datum surface of the surface form stylus profiler \(\theta_{X}\) can be expressed by the following equation:  
\[\theta_{x}=\theta_{\text{Linear\_X}}+\theta_{\text{Holder}} \tag{15}\]  
From Equations (14) and (15), the following equation can be obtained:  
\[\theta_{X}=\theta_{\text{Scan\_Y}}+\theta_{\text{Linear\_X}}-\theta_{\text{ Linear\_Y}} \tag{16}\]  
Since \(\theta_{\text{Linear\_X}}\) and \(\theta_{\text{Linear\_Y}}\) can be treated as the known parameters through scanning the top surface of the linear stage by the surface form stylus profiler in advance of the roundness measurement, \(\theta_{X}\) can be evaluated by measuring \(\theta_{\text{Scan\_Y}}\) in the first step. It should be noted that the parallelism of the cylinder/workpiece with respect to the \(X\)-axis was confirmed visually, and that there should exist a certain amount of angular misalignment of the workpiece about the \(Z\)-axis (\(\varphi_{Z}\)), which could affect the measurement. According to the geometric relationship, the error \(\epsilon_{OX}\) in measurement of \(\theta_{X}\) due to \(\varphi_{Z}\) can be expressed by the following equation:  
\[e_{\theta X}=\theta_{X}-\tan^{-1}(\cos\varphi_{Z}\tan\theta_{X}) \tag{17}\]  
Under the condition of \(\theta_{Z}\) = 2.5\({}^{\circ}\) and \(\theta_{X}\) = 0.1\({}^{\circ}\), \(e_{\theta X}\) becomes approximately 0.002\({}^{\circ}\), and is small enough to be neglected. Meanwhile, the influence of the demounting and remounting operations of the holder unit in measurement of \(\theta_{X}\) should be evaluated in experiments. \(\theta_{Z}\) can be evaluated by obtaining the arc profiles of a small cylinder at different \(Y\)-positions with enough intervals. Figure 10 shows a schematic of the evaluation of \(\theta_{Z}\).