## 3 Methodology and Model
### IPSA-DV-Hop Algorithm
#### 3.1.2 Weighted Hyperbolic Strategy Based on Collinearity  
The hyperbolic localization algorithm determines the position of the node to be localized by placing it on a hyperbola, with the beacon node as the focus and the distance between the two beacon nodes as the focal length. The \(d_{ui}\) and \(d_{uj}\) represent the distance from the unknown node \(u\) to the beacon node \(i\) and beacon node \(j\), respectively. The difference between the distances of the two beacon nodes to the unknown node is \(r_{ij}=d_{ui}-d_{uj}\). The coordinates of the nodes are determined from the intersection of the hyperbolas, and the hyperbola equation can be expressed by Equation (14).  
\[r_{ij}+2r_{ij}\cdot d_{uj}+{d_{uj}}^{2}=-2x_{i}\cdot x-2y_{i}\cdot y+{x_{i}}^{2 }+{y_{i}}^{2}+x^{2}+y^{2} \tag{14}\]  
Observations with higher variance in the error term are assigned smaller weights to create a model without heteroskedasticity. The performance index can then be expressed as Equation (15):  
\[\sigma(c)=[b-Ac]^{T}W[b-Ac] \tag{15}\]  
where \(\sigma(c)\) represents the sum of squares of deviations, and \(W\) is a weighted positive definite matrix.  
The estimator obtained from Equation (15) minimizes and satisfies the conditions for the optimal linear unbiased estimate. The coordinates at the extremes can be obtained by taking the partial derivative and setting it equal to zero. The estimation error at this point is  
\[E\bigg{\{}[c-c_{LSW}][c-c_{LSW}]^{T}\big{\}}=(AWA)^{-1}A^{T}WR_{5}WH\Big{(}A^{T }WA\Big{)}^{-1} \tag{16}\]  
According to the Cauchy--Schwartz inequality, we can obtain the following:  
\[E\bigg{\{}\xi_{i}^{T}\bigg{\}}=diag(E\{\xi_{1}\xi_{1}\},E\{\xi_{2}\xi_{2}\}, \cdots,E\{\xi_{n}\xi_{n}\})+B \tag{17}\]  
When the unknown node \(u\) is not collinear with nodes \(i\) and \(j\), the distance error between \(u\) and \(i\), and the distance error between \(u\) and \(j\), are independent of each other. According to the network distribution characteristics, the product of the variance in the error term and the corresponding weight coefficient is a constant that is independent of \(i\). In this case, the heteroskedasticity more approximately satisfies the conditions for the best linear unbiased estimation.  
\[E\Big{\{}w_{i}\cdot\xi_{i}^{2}\Big{\}}=4w_{i}\cdot\textit{Hopsize}_{u}^{2} \cdot h_{ui}^{2}\cdot E\Big{\{}\Delta d_{ui}^{2}\Big{\}}=4 Hopsize_{u}^{2} \cdot\sigma^{2} \tag{18}\]  
When the nodes are collinear, the matrix approaches a singular matrix due to the limited measurement information. The degree of collinearity (DCL) is introduced in the proposed algorithm, with values closer to 1 indicating that the group of nodes is closer to collinearity [19]. The absolute value of the cosine of the angle in the triangle is calculated using Equation (19):  
\[C_{a}=\Bigg{|}\frac{\left({L_{a}}^{2}+{L_{c}}^{2}-{L_{a}}^{2}\right)}{2L_{b} \cdot L_{c}}\Bigg{|},\ C_{b}=\Bigg{|}\frac{\left({L_{a}}^{2}+{L_{c}}^{2}-{L_{ b}}^{2}\right)}{2L_{a}\cdot L_{c}}\Bigg{|},\ C_  
then performs weighted processing on the \(m\) sets of estimated coordinates, with the result serving as the final coordinates.  
\[D_{i}=\frac{\frac{1}{DCL_{i}}}{\sum_{i=1}^{m}\frac{1}{DCL_{i}}},\ W_{i}=\frac{ DCL_{i}+w_{i}}{\sum_{i=1}^{m}(DCL_{i}+w_{i})} \tag{20}\]