## 3 Machine Tool Compensation Assessment
### Compensation Assessment
#### 3.3.2 Sequential Multilateration Measurement Using a TL  
From an experimental point of view, data acquisition is easier with the TL than with the LI. The measurement process is faster but data analysis is more complex. The coordinates in the machine tool reference frame of \(N\) points M\({}_{i}\) (\(x_{i}\), \(y_{i}\), \(z_{i}\)), with \(i\) = 1 to \(N\), are estimated using the multilateration technique. For this purpose, a set of four lengths \(L_{ij}\) is defined for each point M\({}_{i}\), corresponding to four positions \(j\) of the TL. Equation (4) defines the mathematical relationship between the measured length \(L_{ij}\), the coordinates of the different TL positions, the four respective dead-zones (DZ\({}_{j}\)) and the coordinates of the points M\({}_{i}\). TL\({}_{j}\) (\(x_{TLj}\), \(y_{TLj}\), \(z_{TLj}\)) is the TL location for position \(j\).  
\[e_{ij}=L_{ij}+DZ_{j}-\sqrt{\left(x_{i}-x_{TLj}\right)^{2}+\left(y_{i}-y_{TLj} \right)^{2}+\left(z_{i}-z_{TLj}\right)^{2}}. \tag{4}\]  
Then, a non-linear least squares method is used to determine the four TL positions and the four respective dead-zones. This consists in minimizing the sum of all 4.\(N\) squared errors \(e_{ij}\). Afterwards, the set of coordinates of the points M\({}_{i}\) (\(x_{i}\), \(y_{i}\), \(z_{i}\)) is estimated. The residual error vector PM\({}_{i}\) is thus  
Figure 7: Laser interferometer (LI) measurement.  
The response surfaces are defined by a least squares optimization, and their associated statistical confidence boundaries (SCBs) are computed and plotted. The SCB is the theoretical surface envelop derived from the covariance matrix of the set of coefficients b, as deduced from the least squares residues, and calculated for a given risk \(\alpha\). In this study, 5% is the value chosen for \(\alpha\), which is commonly accepted in the metrological field. This leads to a coverage factor s that is rounded to two. As already pointed out, the best estimators \(\mathbf{\hat{B}}_{k}\) of the two sets of experimental responses \(Y_{k}\) are determined by a least squares optimization, using the pseudo inverse method. Subsequently, the error bars U(\(\hat{Y}_{k}\)) of the estimated responses \(\hat{Y}_{k}\) are determined. They are derived from the least squares residues. Their computation is shown in Equation (6). For that purpose, the standard deviations \(\sigma_{Ek}\) are deduced as shown by Equation (7). \(N_{e}\) and \(N_{f}\) respectively represent the number of experiments and the number of factors. For each response, \(Y_{k}\) and any experiment l, the residue \(E_{lk}\) is computed and its standard deviation \(\sigma_{Ek}\) is then derived. Afterwards, \(\sigma_{Ek}\) is propagated to obtain the variance-covariance matrix of \(\mathbf{\hat{B}}_{k}\) (\(\mathbf{var}\mathbf{\hat{B}}_{k}\)). Finally, the Jacobian of the quadratic models is used to propagate the variance-covariance matrices of \(\mathbf{\hat{B}}_{k}\) to obtain the error bars U(\(\hat{Y}_{k}\)), shown in Equation (8). The SCB surfaces are plotted using the set of error bars U(\(\hat{Y}_{k}\)).  
\[E_{k}=Y_{k}-\mathbf{\hat{B}}_{k}\mathbf{\cdot}\mathbf{X}; \tag{6}\]  
\[\sigma_{Ek}=\sqrt{\frac{1}{N_{e}-N_{f}}\cdot\sum\limits_{l=1}^{N_{e}}E_{lk}^{2 }}; \tag{7}\]  
\[\text{U}(\hat{Y}_{k})=s.\sqrt{\mathbf{J}.\mathbf{var}\mathbf{B}_{k}\mathbf{ \cdot}\mathbf{J}^{\intercal}},\text{with }s=2. \tag{8}\]  
The two sets of optimized setup parameters can finally be determined. For that purpose, \(X_{\text{mopt}}\) (\(m=1\) to \(3\)) are obtained by minimizing the responses as shown by Equation (9):  
\[Y_{\text{kopt}}=\min_{Xm}(\hat{Y}_{k}). \tag{9}\]