# Optimal Configuration and Path Planning for UAV Swarms Using a Novel Localization Approach
## 3 Optimal Configuration Analysis
### Static Emitter Scenario  
The relative sensor-emitter geometry is closely related to the location accuracy, which can be reflected by CRLB, and the configuration corresponding to the minimum CRLB is the optimal configuration. For unbiased estimator \(\hat{\mathbf{x}}\) of \(\mathbf{x}\), its Cramer-Rao bound can be expressed as:  
\[E\Big{[}(\mathbf{x}-\hat{\mathbf{x}})(\mathbf{x}-\hat{\mathbf{x}})^{T}\Big{]} \geq\mathbf{J}^{-1}\triangleq\mathbf{CRLB}(\mathbf{x}), \tag{7}\]where **J** is the Fisher information matrix (FIM). Then the FIM for TDOA localization with distance-dependent noise is given by [25]  
\[\textbf{J}_{i,j}=\left[\frac{\partial}{\partial x_{i}}\ln(f_{\textbf{z}}(\hat{ \textbf{z}};\textbf{x}))\frac{\partial}{\partial x_{j}}\ln(f_{\textbf{z}}(\hat{ \textbf{z}};\textbf{x}))\right], \tag{8}\]  
where \(i,j\in\{1,2\}\). This FIM can be divided into two parts; as for the first part,  
\[\textbf{J}_{1,(i,j)}=\frac{\partial\textbf{z}(\textbf{x})}{\partial x_{i}} \boldsymbol{\Sigma}_{r1}{}^{-1}(\textbf{x})\Bigg{(}\frac{\partial\textbf{z}( \textbf{x})}{\partial x_{j}}\Bigg{)}^{T}. \tag{9}\]  
The Jacobian matrix of the measurement set with receiver \(1\) as the reference receiver is  
\[ \begin{array}{l}\frac{\partial\textbf{z}(\textbf{x})}{\partial x_{1}}= \left[\frac{\partial x_{12}(\textbf{x})}{\partial x_{1}},\frac{\partial x_{1 3}(\textbf{x})}{\partial x_{1}},\cdots,\frac{\partial x_{1M}(\textbf{x})}{ \partial x_{1}}\right]^{T}\\ =[\cos(\theta_{2})-\cos(\theta_{1}),\cos(\theta_{3})-\cos(\theta_{1}),\cdots, \cos(\theta_{M})-\cos(\theta_{1})]^{T}\end{array}  \tag{10}\]  
\[\frac{\partial\textbf{z}(\textbf{x})}{\partial x_{2}}=[\sin(\theta_{2})-\sin( \theta_{1}),\sin(\theta_{3})-\sin(\theta_{1}),\cdots,\sin(\theta_{M})-\sin( \theta_{1})]^{T}, \tag{11}\]  
with \(\theta_{i}\) is the angle of arrival measurement of the \(i\)-th sensor and the emitter. For the second part, when \(r_{i}>r_{0}\),  
\[\textbf{J}_{2,(i,j)}=\frac{1}{2}Tr\Bigg{(}\boldsymbol{\Sigma}_{r1}{}^{-1}( \textbf{x})\frac{\partial\boldsymbol{\Sigma}_{r1}(\textbf{x})}{\partial x_{i} }\boldsymbol{\Sigma}_{r1}{}^{-1}(\textbf{x})\frac{\partial\boldsymbol{\Sigma} _{r1}(\textbf{x})}{\partial x_{j}}\Bigg{)}, \tag{12}\]  
where the Jacobian matrix for computing the distance dependent FIM is expressed by  
\[\frac{\partial\boldsymbol{\Sigma}_{r1}(\textbf{x})}{\partial x_{1}}=2\beta \left[ \begin{array}{cccc}r_{1}\cos\theta_{1}+r_{2}\cos\theta_{2}&r_{1}\cos \theta_{1}&\cdots&r_{1}\cos\theta_{1}\\ r_{1}\cos\theta_{1}&r_{1}\cos\theta_{1}+r_{3}\cos\theta_{3}&\cdots&r_{1}\cos \theta_{1}\\ \vdots&\vdots&\ddots&\vdots\\ r_{1}\cos\theta_{1}&r_{1}\cos\theta_{1}&\cdots&r_{1}\cos\theta_{1}+r_{(M-1)} \cos\theta_{(M-1)}\end{array} \right] \tag{13}\]  
\[\frac{\partial\boldsymbol{\Sigma}_{r1}(\textbf{x})}{\partial x_{2}}=2\beta \left[ \begin{array}{cccc}r_{1}\sin\theta_{1}+r_{2}\sin\theta_{2}&r_{1}\sin \theta_{1}&\cdots&r_{1}\sin\theta_{1}\\ r_{1}\sin\theta_{1}&r_{1}\sin\theta_{1}+r_{3}\sin\theta_{3}&\cdots&r_{1}\sin \theta_{1}\\ \vdots&\vdots&\ddots&\vdots\\ r_{1}\sin\theta_{1}&r_{1}\sin\theta_{1}&\cdots&r_{1}\sin\theta_{1}+r_{(M-1)} \sin\theta_{(M-1)}\end{array} \right], \tag{14}\]  
where \(\beta=\frac{a}{SNR_{0}r_{0}^{2}}\). Based on the characteristics of the FIM, the optimal configuration is analyzed via distance rule and angle rule. (1) Distance rule  
As pointed out in [26], arbitrarily selecting a reference sensor does not change the CRLB for TDOA-based source localization with distance-independent noises. Here, we extend it to the distance-dependent noise model. **Theorem 1**. : _Given the positions of the receivers and emitter, i.e., given distance \(r_{i}\) and angle \(\theta_{i}\) the election of reference receiver has no impact on the CRLB with distance-dependent noise._  
**Proof. ** Without loss of generality, receivers 1 and 2 are taken as the reference receivers. Then the TDOA measurement with different reference receivers can be represented by  
\[\hat{\mathbf{z}}_{r1}=\left[