## 3 Main Results
### Communication Interference Resistance with Sensors
: _Extra reference can be transmitted among agents or obtained by detecting beside bearings, such as distance or velocity, then we can obtain \(\mathbf{X}_{ij}^{t}\) with Equations (37) and (38) if \(rank\big{(}\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\big{)}=N-\text{N}_{l}\) holds._  
**Situation 3**. : _Although extra reference can be transmitted among agents or obtained by detecting besides bearings, systems encounter communication breaks and some agents lose interaction with others, then we can obtain \(\mathbf{X}_{ij}^{t}\) with Equations (37) and (38) if \(rank\big{(}\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\big{)}=N-\text{N}_{l}\) holds, and the number of positive elements of the corresponding rows of \(\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\) must be greater than \(n\) for agents without any interaction._  
According to the discussion, formation reconstruction might be necessary while encountering communication breaks. The autonomous system should adjust the motion parameters, thus agents without interaction could obtain sufficient measurement. With the above strategy, the estimator is proposed as:  
\[\left(A_{s}^{i}\otimes\mathbf{1}_{1\times 2n}\right)^{\text{T}}\circ H\Big{(}\mathbf{ X}_{ij}^{t}\Big{)}=\mathbf{0} \tag{39}\]  
where \(\mathbf{A}_{s}^{i}=\left[a_{s}^{ij}\right]_{1}^{N}\) represents the adjacent vector of agent \(i\), \(\mathbf{H}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}=\left[\mathbf{h}\Big{(}\mathbf{X}_{ij}^{t}\Big{)} \right]_{2 nN}^{1}\), and \(\mathbf{h}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}\) is indicated as:  
\[\mathbf{h}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}= \begin{bmatrix}\mathbf{b}_{ij}^{t}\|\mathbf{X}_{ij }^{t}\|_{2}-\mathbf{X}_{ij}^{t}\\ D_{1}\Big{(}\mathbf{b}_{ij}^{t}\Big{)}-\frac{D_{1}\big{(}\mathbf{X}_{ij}^{t}\big{)}}{\| \mathbf{X}_{ij}^{t}\|_{2}}+\frac{\mathbf{X}_{ij}^{t}\big{(}\mathbf{X}_{ij}^{t}\big{)}^{ \text{T}}D_{1}\big{(}\mathbf{X}_{ij}^{t}\big{)}}{\|\mathbf{X}_{ij}^{t}\|_{2}^{2}}\end{bmatrix}  \tag{40}\]  
Considering the nonlinearity and measured deviation, (38) might have no closed-form analytic solution. Therefore, the nonlinear programming method is applied here, and the corresponding regulator is defined as:  
\[ \begin{split} J&=\big{\|}\left(\mathbf{A}_{s}^{i}\otimes \mathbf{1}_{1\times n}\right)^{\text{T}}\circ\mathbf{G}_{\mathbf{X}}\big{\|}_{\mathbf{Q}_{s}} \\ s.t.&\quad\quad\left(\mathbf{A}_{s}^{i}\otimes\mathbf{1}_{1 \times n}\right)^{\text{T}}\circ\mathbf{G}_{\mathbf{b}}=\mathbf{0}\end{split}  \tag{41}\]  
where \(\mathbf{Q}_{s}\) is a symmetric positive matrix. \[\left\{ \begin{array}{l}\mathbf{G}_{\mathbf{b}}=\left[\mathbf{b}_{ij}^{t}\|\mathbf{X}_{ij}^{t }\|_{2}-\mathbf{X}_{ij}^{t}\right]_{nN}^{1}\\ \mathbf{G}_{\mathbf{X}}=\Bigg{[}D_{1}\Big{(}\mathbf{b}_{ij}^{t}\Big{)}-\frac{D_{1}\big{(} \mathbf{X}_{ij}^{t}\big{)}}{\|\mathbf{X}_{ij}^{t}\|_{2}}+\frac{\mathbf{x}_{ij}^{t}\big{(} \mathbf{X}_{ij}^{t}\big{)}^{\text{T}}D_{1}\big{(}\mathbf{X}_{ij}^{t}\big{)}}{\|\mathbf{X}_{ ij}^{t}\|_{2}^{2}}\Bigg{]}_{nN}^{1}\end{array} \right. \tag{42}\]  
For agents that can keep communication with neighbors, the sensing information can be used for unbiased estimations. For agents without interaction (encountering communication breaks), sensing information is crucial for consensus keeping.