## 3 Our Approach to Estimation of Robot Pose and Pipe Diameter (EPD)
### Analysis of Residual Asymmetry in C-EPD
On the other hand, by substituting (14) and (15), the first term in the RHS of (13) is simplified as follows:  
\[r_{c}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})+r_{c}(\mathbf{x}_{i,S}- \mathbf{e}_{i,S},\mathbf{\beta})=\\ 2\Bigg{(}\frac{y_{i,P}^{2}}{(D_{\text{max}}/2)^{2}}+\frac{z_{i, P}^{2}}{(D_{\text{min}}/2)^{2}}-1\Bigg{)}+\Bigg{(}\frac{\mathbf{e}_{y,i,P}^{2}}{(D_{ \text{max}}/2)^{2}}+\frac{\mathbf{e}_{z,i,P}^{2}}{(D_{\text{min}}/2)^{2}}\Bigg{)} \tag{16}\]  
Note that the first term in the RHS of (16) equals zero at the ground-truth point, while the second term is always positive due to the squared terms. Since both terms in (13) are positive, the expanded inequality \(r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})>r_{c}^{2}(\mathbf{x}_{i,S}-\mathbf{e}_ {i,S},\mathbf{\beta})\) proves that positive errors incur larger residuals than negative errors, causing systematic overestimations in pipe diameter estimation. There is a relationship between pipe diameter, random depth error, and the estimation bias in the canonical ellipse residual function that can be inferred from (16). As the random depth error increases, the second term in (16) becomes larger. This behavior amplifies the asymmetry in \(r_{c}(\mathbf{\hat{x}}_{i,S},\mathbf{\beta})\), resulting in a systematic estimation bias. Due to this asymmetry, the canonical ellipse residual function causes the optimization process to estimate larger diameters than the ground-truth values. Theorem 1 formally states that the optimal solution \(\mathbf{\beta}_{\text{C-}EPD}^{*}\) in (7) produces overestimated pipe diameters \((\hat{D}_{\text{max}},\hat{D}_{\text{min}})\) compared to the ground-truth diameters \((D_{\text{max}},D_{\text{min}})\). Theorem 1 (Diameter Estimation Bias). : _Given zero-mean Gaussian random depth errors and the canonical ellipse residual function, the estimated diameters are systematically overestimated, i.e., \(\hat{D}_{\text{max}}>D_{\text{max}}\) and \(\hat{D}_{\text{min}}>D_{\text{min}}\)._Proof. : At the optimal diameters \((\hat{D}\text{max},\hat{D}\text{min})\), the objective function \(F(\mathbf{\beta})\) in (4) satisfies the following conditions:  
\[\frac{\partial F}{\partial\hat{D}_{\text{max}}}=0\quad\text{and}\quad\frac{ \partial F}{\partial\hat{D}_{\text{min}}}=0. \tag{17}\]  
From (14) and (15), the partial derivatives with respect to \(\hat{D}_{\text{max}}\) and \(\hat{D}_{\text{min}}\) are computed by  
\[\pi r_{c}(\tilde{x}_{i,P},\mathbf{\beta})=\pi(\tilde{y}_{i,P}^{2}+\tilde{z}_{i,P}^ {2})-\pi, \tag{11}\]0  
and  
\[\pi r_{c}(\tilde{x}_{i,P},\mathbf{\beta})=\pi(\tilde{y}_{i,P}^{2}+\tilde{z}_{i,P}^ {2})-\pi, \tag{11}\]1  
respectively. We observe that the terms \(\big{(}y_{i,P}+\epsilon_{y,i,P}\big{)}^{2}\) and \((z_{i,P}+\epsilon_{z,i,P})^{2}\) for positive-error points dominate over those for negative-error points due to their squared values. Furthermore, from Lemma 1, the residual function \(r_{c}(\mathbf{x}_{i,S}+\mathbf{\epsilon}_{i,S},\mathbf{\beta})\) associated with positive-error points is greater in magnitude than that of negative-error points. Given that the random depth errors are modeled as zero-mean Gaussian distributions, the contributions from positive-error points dominate the optimization process. As a result, the optimizer compensates for the larger residuals by increasing the diameters \(\hat{D}_{\text{max}}\) and \(\hat{D}_{\text{min}}\) to reduce their impact, leading to \(\hat{D}_{\text{max}}>D_{\text{max}}\) and \(\hat{D}_{\text{min}}>D_{\text{min}}\). This systematic adjustment confirms the estimation bias in the canonical ellipse residual function.