# Resilient Formation Reconfiguration for Leader-Follower Multi-UAVs
## 4 Construction of the Hierarchical BRB Performance Assessment Model for UAVs
### Inference Process Description  
Inside \(BRB_{11}\) in the basic layer, the \(f\)th antecedent attribute, \(f=1,2\), will be transformed into the following belief distribution:  
\[\{(A_{11,f}^{1},\alpha_{11,f}^{1}(t)),\ldots,(A_{11,f}^{I_{1,f}},\alpha_{11,f }^{I_{1,f}}(t))\}. \tag{19}\]  
Therein, \(\alpha_{11,f}^{j}(t)\), \(f=1,2\), \(j=1,\ldots,I_{1,f}\), is the belief degree of the \(j\)th referential value with respect to the \(f\)th antecedent attribute of \(BRB_{11}\) in the basic layer, which can be calculated according to the flowchart displayed in Figure 3.  
Denote \(\alpha_{11,f}(t)=[a^{1}_{11,f}(t),\ldots,a^{I_{11,f}}_{11,f}(t)]^{T},f=1,2\), and then, the antecedent attribute matching degree matrix \(M_{11,a}\in\mathbb{R}^{L_{11}\times 2}\) of \(X_{11}(t)\) and \(X_{12}(t)\) can be obtained by  
\[ \begin{split} M_{11,a}(t)&=[M_{11,a1}(t),M_{11,a2}(t )],\\ M_{11,a1}(t)&=\alpha_{11,1}(t)\otimes\mathbf{1}_{ \mathbf{I}_{11,2}}\\ &=\Big{[}m^{1}_{11,a1}(t),\ldots,m^{L_{11}}_{11,a1}(t)\Big{]}^{T}, \\ M_{11,a2}(t)&=\mathbf{1}_{\mathbf{I}_{11,1}}\otimes \alpha_{11,2}(t)\\ &=\Big{[}m^{1}_{11,a2}(t),\ldots,m^{L_{11}}_{11,a2}(t)\Big{]}^{T}. \end{split}  \tag{20}\]  
Figure 3: Flowchart of calculating the belief degree of the \(j\)th referential value with respect to the \(f\)th antecedent attribute \(a^{j}_{11,f}(t),f=1,2,j=1,\ldots,J_{11,f}\). Therein, \(m^{k}_{11,af}(t)\), \(f=1,2\), \(k=1,\ldots,L_{11}\) denotes the matching degree between the \(f\)th input and the \(f\)th antecedent attribute of the \(k\)th belief rule, \(\otimes\) denotes the Kronecker product of matrices while \(\mathbf{1}_{l1,f}\in\mathbb{R}^{l1_{l},f},f=1,2\), denotes the column vector with all entries \(1\). Subsequently, one can obtain the following rule matching degree matrix:  
\[ \begin{split} M_{11,r}(t)&=\bar{M}_{11,a1}(t) \odot\bar{M}_{11,a2}(t)\\ &=\left[m^{1}_{11,r}(t),\ldots,m^{L_{11}}_{11,r}(t)\right]^{T}, \end{split}  \tag{21}\]  
where \(\odot\) denotes the Hadamard product of matrices and  
\[ \begin{split}\bar{M}_{11,a1}(t)&=\begin{bmatrix}(m^ {1}_{11,a1}(t))^{\delta^{11}_{1}}\\ \vdots\\ (m^{L_{11}}_{11,a1}(t))^{\delta^{11}_{1}}\end{bmatrix},\\ \bar{M}_{11,a2}(t)&=\begin{bmatrix}(m^{1}_{11,a2}(t))^{\delta^{11}_{ 2}}\\ \vdots\\ (m^{L_{11}}_{11,a2}(t))^{\delta^{11}_{2}}\end{bmatrix},\\ \bar{\delta}^{11}_{f}&=\frac{\delta^{11}_{f}}{\max\limits_{f=1,2}\{\delta^{11 }_{f}\}},f=1,2.\end{split}  \tag{22}\]  
Based on the rule matching degree matrix (21), the activation weight of the \(k\)th belief rule of \(BRB_{11}\), \(k=1,\ldots,L_{11}\), can be yielded by  
\[\omega^{11}_{k}(t)=\frac{\theta^{11}_{k}m^{k}_{11,r}(t)}{\sum\limits_{l=1}^{L_ {11}}\theta^{11}_{l}m^{l}_{11,r}(t)}. \tag{23}\]  
Then, all activated belief rules will be fused using the following analytical ER algorithm as [38]:  
\[ \begin{split}\beta^{11}_{p}(t)&=\frac{\mu(t)(\prod \limits_{k=1}^{L_{11}}(\omega^{11}_{k}(t)\beta^{11}_{k,p}+1-\omega^{11}_{k}(t) \sum\limits_{p=1}^{P_{11}}\beta^{11}_{k,p})-\prod\limits_{k=1}^{L_{11}}(1- \omega^{11}_{k}(t)\sum\limits_{p=1}^{P_{11}}\beta^{11}_{k,p})}{1-\mu(t)\prod \limits_{k=1}^{L_{11}}(1-\omega^{11}_{k}(t))},\end{split}  \tag{24}\]  
\[ \begin{split}\mu(t)&=\frac{1}{\sum\limits_{p=1}^{P_ {11}}\prod\limits_{k=1}^{L_{11}}(\omega^{11}_{k}(t)\beta^{11}_{k,p}+1-\omega^{1 1}_{k}(t)\sum\limits_{p=1}^{P_{11}}\beta^{11}_{k,p})-(P_{11}-1)\prod\limits_{k= 1}^{L_{11}}(1-\omega^{11}_{k}(t)\sum\limits_{p=1}^{P_{11}}\beta^{11}_{k,p})}. \end{split}  \tag{25}\]  
Therein, \(\mu(t)\) is the correction factor and \(\beta^{11}_{p}(t)\), \(p=1,\ldots,P_{11}\), denotes the aggregated belief degree of the \(p\)th assessment grade \(D^{11}_{p}\).