# Resilient Formation Reconfiguration for Leader-Follower Multi-UAVs
## 4 Construction of the Hierarchical BRB Performance Assessment Model for UAVs
### Belief Rule Definition  
In terms of \(BRB_{11}\) in the basic layer of the hierarchical BRB performance assessment model for UAVs, the \(k\)th belief rule can be defined as  
\[Rule_{k}^{11}:\] \[If\ X_{11}(t)\ is\ A_{11,k,1}\wedge X_{12}(t)\ is\ A_{11,k,2},\] \[Then\ \{(D_{1}^{11},\beta_{k,1}^{11}),\ldots,(D_{P_{11}}^{11},\beta_{ k,P_{11}}^{11})\}, \tag{12}\] \[With\ rule\ weight\ \theta_{k}^{11},\ antecedent\ attribute\ weights\ \delta_{1}^{11},\delta_{2}^{11}.\]  
Therein, \(X_{11}(t)\) and \(X_{12}(t)\) are the antecedent attributes of \(BRB_{11}\) corresponding to the kinematic pivotal performance metrics \(\mathcal{M}_{11}^{i}(t)\) and \(\mathcal{M}_{12}^{i}(t)\), \(i=1,\ldots,N\), respectively. \(A_{11,k,f}\in A_{11,f},k=1,\ldots,L_{11},f=1,2\) refers to the referential value of the \(f\)th antecedent attribute in the \(k\)th rule. Note that \(A_{11,1,f},\ldots,A_{11,L_{11},f}\) are not different from each other. \(A_{11,f}=\{A_{11,f}^{1},\ldots,A_{11,f}^{11,f}\}\) is the corresponding referential value set with \(J_{11,f}=|A_{11,f}|\), where \(|\cdot|\) denotes the amount of the elements contained in the set and the sequence \(A_{11,f}^{1},\ldots,A_{11,f}^{I_{11},f}\) is assumed to be increasing, which means \(A_{11,f}^{1},\ldots,A_{11,f}^{I_{11},f}\) are different from each other. \(L_{11}\) signifies the amount of the belief rules in \(BRB_{11}\) and is calculated by \(L_{11}=\prod\limits_{f=1}^{2}J_{11,f}\). \(\beta_{k,p}^{11},k=1,\ldots,L_{11}\), \(p=1,\ldots,P_{11}\), is the belief degree of the \(p\)th assessment grade \(D_{p}^{11}\) in the \(k\)th rule. Here, one assumes that each belief rule of \(BRB_{11}\) is complete, that is,  
\[\sum\limits_{p=1}^{P_{11}}\beta_{k,p}^{11}=1,k=1,\ldots,L_{11}, \tag{13}\]  
while the weight of the belief rules and the antecedent attributes are normalized, namely,  
\[ \begin{split} 0\leq\theta_{k}^{11}&\leq 1,k=1, \ldots,L_{11},\\ 0\leq\delta_{f}^{11}&\leq 1,f=1,2.\end{split}  \tag{14}\]  
The \(k\)th belief rule of \(BRB_{12}\) in the basic layer of the hierarchical BRB performance assessment model for UAVs can be represented as  
\[ \begin{split} Rule_{k}^{12}:\\ If\ X_{21}(t)\ is\ A_{12,k,1}\wedge X_{22}\ is\ A_{12,k,2},\\ Then\ \{(D_{1}^{12},\beta_{k,1}^{12}),\ldots,(D_{P_{12}}^{12},\beta_{ k,P_{12}}^{12})\},\\ With\ rule\ weight\ \theta_{k}^{12},\ antecedent\ attribute\ weights\ \delta_{1}^{12},\delta_{2}^{12},\end{split}  \tag{15}\]  
where \(X_{21}(t)\) and \(X_{22}\) are the antecedent attributes of \(BRB_{12}\) corresponding to the non-kinematic pivotal performance metrics \(\mathcal{M}_{21}^{i}(t)\) and \(\mathcal{M}_{22}^{i}\), \(i=1,\ldots,N\), respectively. \(A_{12,k,f}\in A_{12,f},k=1,\ldots,L_{12},f=1,2\), is the referential value of the \(f\)th antecedent attribute in the \(k\)th rule of \(BRB_{12}\) while \(A_{12,f}=\{A_{12,f}^{1},\ldots,A_{12,f}^{I_{12},f}\}\) is the corresponding referential value set with \(J_{12,f}=|A_{12,f}|\). It is supposed that the sequence \(A_{12,f}^{1},\ldots,A_{12,f}^{I_{12,f}}\) is increasing, that is, \(A_{12,f}^{1},\ldots,A_{12,f}^{I_{12,f}}\) are different from each other. By contrast, \(A_{12,1,f},\ldots,A_{12,L_{12},f}\) are not different from each other. \(\beta_{k,p}^{12},k=1,\ldots,L_{12},p=1,\ldots,P_{12}\), represents the belief degree of the \(p\)th assessment grade \(D_{p}^{12}\) in the \(k\)th rule.