## 3 Experiments and Results
### Calibration and Measurement  
To measure the length of a standard gauge block using the single joint setup, joint parameters must be calibrated. The calibration and measurement model are illustrated in Figure 10. In this figure, \(R\) is the radius of the cantilever, and \(r\) is the radius of the probe head. \(d_{1}\) and \(d_{2}\) represent the standard dimensions of two employed gauge blocks. \(\theta_{1}\) and \(\theta_{2}\) represent the angle values measured by the encoder when the probe touches each side of the gauge block, respectively. \(L_{1}\) and \(L_{2}\) are installation structural parameters for the device and \(\alpha\) is the angle formed by \(L_{1}\) and \(R\). If only the heights of the two blocks are measured, these dimensions have the relationship shown in Equation (4). The parameters of this model can be calibrated by using standard gauge blocks of different heights.  
\[\left\{\begin{array}{l}\cos(a)=\frac{L_{1}+r}{R};\\ \sin(a)=\frac{L_{2}}{R};\\ \cos(a+\theta_{1})=\cos(a)\cos(\theta_{1})-\sin(a)\sin(\theta_{1})=\frac{L_{ 1}-d_{1}-r}{R};\\ \cos(a+\theta_{2})=\cos(a)\cos(\theta_{2})-\sin(a)\sin(\theta_{2})=\frac{L_{ 1}-d_{2}-r}{R};\end{array}\right. \tag{4}\]  
The optimal parameters for the model can be identified through calibration. The device was used to measure a number of gauge blocks of known sizes (\(d_{1}\),..., \(d_{n}\)) to collect multiple angle data (\(\theta_{1}\),..., \(\theta_{n}\)). In this manner, the over-determined expressions in Equation (5) were obtained. The least-squares solution for this equation was solved to obtain \(L_{1}\), \(L_{2}\), and \(R\).  
\[\left[\begin{array}{cc}1-\cos\theta_{1}&\sin\theta_{1}\\ 1-\cos\theta_{2}&\sin\theta_{2}\\ \cdots&\cdots\\ 1-\cos\theta_{n}&\sin\theta_{n}\end{array}\right]\left[\begin{array}{c}L_{ 1}\\ L_{2}\end{array}\right]=\left[\begin{array}{c}r(1+\cos\theta_{1})+d_{1}\\ r(1+\cos\theta_{2})+d_{2}\\ \cdots\\ r(1+\cos\theta_{n})+d_{n}\end{array}\right] \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**PWM/Steady-State** & **4\%/0.23** & **6\%/0.67** & **8\%/1.10** & **10\%/1.53** & **12\%/1.97** & **14\%/2.27** \\ \hline \(\sigma_{\text{U}}\) & 0 & 0.85 & 0 & 1.01 & 1.21 & 0 \\ \(\sigma_{\text{D}}\) & 0.64 & 0.85 & 1.1 & 0 & 0 & 1.8 \\ \hline \end{tabular}
\end{table}
Table 5: Variance distributions at different speeds.  
Figure 9: Residual error in count values (speed = 0.67 rad/s).  
Zero-level gauge blocks with heights of 7 mm, 7.5 mm, 8 mm, 8.5 mm, 9 mm, 9.5 mm, and 10 mm were installed. The motor rotated at a speed of 0.23 rad/s (PWM = 4%). When the probe touched the upper and lower surfaces of the gauge blocks, the encoder count values were collected, and \(\theta_{\mathrm{i}}\) was used to denote their difference. The experimental data are listed in Table 6. The measurement data from the 7 mm, 8 mm, 8.5 mm, 9 mm, and 10 mm blocks were used to calibrate the model parameters. The least-squares solutions are calculated \(L_{1}\) = 96.5933 mm, \(L_{2}\) = 248.6310 mm, and \(R\) = 266.7351 mm. After determining the optimal model parameters, the size of a measured gauge block can be calculated using Equation (6) according to the measured values.  
\[d=L_{1}[1-\cos(\theta)]+L_{2}\sin(\theta)-r[1+\cos(\theta)] \tag{6}\]  
By taking the blocks with heights of 9.5 mm and 7.5 mm as components to be tested and substituting the measured values of \(\theta_{\mathrm{i}}\) into Equation (6), the results of 9.5148 mm and 7.4976 mm were obtained, representing measurement errors of \(-\)14.8 \(\upmu\)m and 2.4 \(\upmu\)m, respectively.  
The results of the two experiments show that the proposed design is suitable for the self-driven AACMM. The measurement errors mainly come from the identification of structural parameters. In addition, the angle measuring error, the deformation error of the probe bar, and the gross error in the measuring process are also primary error sources.