## 3 Our Approach to Estimation of Robot Pose and Pipe Diameter (EPD)
### Analysis of Residual Asymmetry in C-EPD
The residual \(\pi r_{c}(\tilde{x}_{i,P},\mathbf{\beta})\) corresponds to the shaded annular area between these two circles, as shown in Figure 2. This figure illustrates that the residual area for positive-error points (red region) is larger than that for negative-error points (green region) given the same magnitude of error, \(\mathbf{\epsilon}_{i}\). This asymmetry causes a systematic estimation bias during optimization, favoring larger diameters to minimize errors. Lemma 1 formalizes this observation by proving that the canonical residual function systematically assigns larger errors to positive-error points compared to negative-error points with identical error magnitudes. **Lemma 1** (Asymmetry in Canonical Residual Function). : _For two measurement points, \(\mathbf{x}_{i,S}+\mathbf{\epsilon}_{i,S}\) and \(\mathbf{x}_{i,S}-\mathbf{\epsilon}_{i,S}\), where \(|\mathbf{x}_{i,S}+\mathbf{\epsilon}_{i,S}|>|\mathbf{x}|>|\mathbf{x}_{i,S}-\mathbf{\epsilon}_{i,S}|\), the squared canonical residual function satisfies_  
\[r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{\epsilon}_{i,S},\mathbf{\beta})>r_{c}^{2}(\mathbf{x}_{i,S}- \mathbf{\epsilon}_{i,S},\mathbf{\beta}). \tag{12}\]  
Figure 2: Conceptual representation of canonical ellipse residual function in coordinate system scaled by \(D_{\text{max}}\) and \(D_{\text{min}}\). _This result demonstrates that the canonical residual function penalizes positive errors more heavily than negative errors, leading to systematic overestimations of pipe diameters in (7)._  
Proof. : To prove \(r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})>r_{c}^{2}(\mathbf{x}_{i,S}-\mathbf{e}_{i, S},\mathbf{\beta})\), we begin by evaluating the difference between the squared residuals: \(r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})-r_{c}^{2}(\mathbf{x}_{i,S}-\mathbf{e}_{ i,S},\mathbf{\beta})>0\). Then, we expand \(r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})-r_{c}^{2}(\mathbf{x}_{i,S}-\mathbf{e}_{ i,S},\mathbf{\beta})\) as follows:  
\[r_{c}^{2}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})-r_{c}^{2}(\mathbf{x}_ {i,S}-\mathbf{e}_{i,S},\mathbf{\beta})=\\ [r_{c}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})+r_{c}(\mathbf{x}_{i,S}- \mathbf{e}_{i,S},\mathbf{\beta})]\cdot[r_{c}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})-r_{ c}(\mathbf{x}_{i,S}-\mathbf{e}_{i,S},\mathbf{\beta})] \tag{13}\]  
Denoting a transformed vector of \(\mathbf{e}_{i,S}\) in the PCS by \(\mathbf{e}_{i,P}=[\mathbf{e}_{x,i,P},\mathbf{e}_{y,i,P},\mathbf{e}_{z,i,P}]^{T}\), the canonical residual functions for the positive and negative errors are expressed as  
\[r_{c}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})=\frac{(y_{i,P}+\mathbf{e}_{y,i,P})^{2} }{(D_{\text{max}}/2)^{2}}+\frac{(z_{i,P}+\mathbf{e}_{z,i,P})^{2}}{(D_{\text{min}}/ 2)^{2}}-1, \tag{14}\]  
and  
\[r_{c}(\mathbf{x}_{i,S}-\mathbf{e}_{i,S},\mathbf{\beta})=\frac{(y_{i,P}-\mathbf{e}_{y,i,P})^{2} }{(D_{\text{max}}/2)^{2}}+\frac{(z_{i,P}-\mathbf{e}_{z,i,P})^{2}}{(D_{\text{min}}/ 2)^{2}}-1, \tag{15}\]  
respectively. From geometry, the positive-error point \(\mathbf{x}_{i,S}+\mathbf{e}_{i,S}\) lies outside the pipe, making \(r_{c}(\mathbf{x}_{i,S}+\mathbf{e}_{i,S},\mathbf{\beta})>0\). Conversely, the negative-error point \(\mathbf{x}_{i,S}-\mathbf{e}_{i,S}\) lies inside the pipe, yielding \(r_{c}(\mathbf{x}_{i,S}-\mathbf{e}_{i,S},\mathbf{\beta})<0\). Consequently, the second term in the right-hand side (RHS) of (13) becomes positive, contributing to larger residuals for positive errors.