## 3 Main Results
### Communication Interference Resistance with Sensors  
Based on Lemma 2, the nonlinear controller proposed in Section 3.3 might become invalid when agents encounter communication breaks as conditions of Proposition 1 cannot be met. Therefore, extra reference information should be provided, and sensors mounted on agents can complete the mission. In this paper, we discuss the requirements of the sensing graph and propose an estimator with only bearings measurement. Firstly, we introduce the sensing graph \(\mathcal{G}_{s}\), which is indicated as \((\mathcal{V}_{s},\mathcal{E}_{s})\), where \(\mathcal{V}_{s}=\big{\{}v_{s}^{i}|i\in\mathcal{N}_{a}\big{\}}\) indicates the vertices, and \(\mathcal{E}_{s}\subseteq\mathcal{V}_{s}\times\mathcal{V}_{s}\) represents the edges. Similar to \(\mathcal{G}\), the adjacent matrix of \(\mathcal{G}_{s}\) is defined as \(\mathbf{A}_{s}=\left[\dot{a}_{s}^{ij}\right]_{N}\), \(a_{s}^{ij}>0\) if agent \(i\) can sense information of agent \(j\), otherwise \(a_{ij}=0\). Obviously, \(\mathbf{A}_{s}\) is not necessary to be symmetric. Subsequently, the Laplacian matrix \(\mathbf{\mathcal{L}}_{s}\) is indicated as \(\mathbf{\mathcal{L}}_{s}=\left[l_{s}^{ij}\right]_{N}^{N}\), \(l_{s}^{ij}=a_{s}^{ij}\) if \(i\
eq j\), otherwise \(l_{s}^{ii}=-\sum a_{s}^{ij}\). Stimulated by [22] restrictions should be put to \(\mathbf{\mathcal{L}}_{s}\) with insufficient measured information, otherwise, the proposed controller in Proposition 1 will be invalid. The primary restriction is given with Theorem 2. **Theorem 2**. : _Multi-agent systems can keep stable with \(\text{rank}\big{(}\mathbf{\mathcal{L}}\big{)}<N-N_{l}\) and \(\text{rank}\big{(}\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\big{)}=N-N_{l}\), if there exist communication interferences._  
Theorem 2 can be proved with Lemma 1 and various restrictions for different sensors. Next, we will discuss these restrictions required for stable formation. As seen from the reference information processing in Section 2.3, distance information [8] is necessary for formation control, even in [1; 21; 25; 29], velocity measurement is required for controller design. Therefore, bearings alone cannot provide sufficient information without information fusion. In this paper, we propose an estimator for relative positions with bearings only, and we give the model of bearings \(\mathbf{b}_{ij}^{t}\)\(\forall(i,j)\in\mathcal{E}_{s}\):  
\[\mathbf{b}_{ij}^{t}=\frac{\mathbf{X}_{ij}^{t}}{\|\mathbf{X}_{ij}^{t}\|_{2}} \tag{37}\]  
where \(\mathbf{X}_{ij}^{t}=\mathbf{\mathcal{T}}_{0i}\Big{(}\mathbf{X}_{t}^{i}-\mathbf{X}_{t}^{i} \Big{)}\). Then, take derivatives of \(\mathbf{b}_{ij}^{t}\) and we can obtain:  
\[D_{1}\Big{(}\mathbf{b}_{ij}^{t}\Big{)}=\frac{D_{1}\Big{(}\mathbf{X}_{ij}^{t}\Big{)} \|\mathbf{X}_{ij}^{t}\|_{2}^{2}-\mathbf{X}_{ij}^{t}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}^{ \mathsf{T}}D_{1}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}}{\|\mathbf{X}_{ij}^{t}\|_{2}^{3}} \tag{38}\]  
**Remark 3**. : _Equations (37) and (38) are both underdetermined with bearing \(\mathbf{b}_{ij}^{t}\) only. If the relative positions can be obtained [8], then \(\mathbf{b}_{ij}^{t}\) is useless. If the velocity can be sensed [21], then Equation (38) is determined and \(D_{1}\Big{(}\mathbf{X}_{ij}^{t}\Big{)}\) can be obtained. The above analysis implies that the controller is valid with bearings information if Equations (37) and (38) can be solved, which means we must obtain \(\mathbf{X}_{ij}^{t}\) with Equations (37) and (38), which can be considered as restrictions on \(\mathbf{\mathcal{L}}_{s}\) in various situations._  
**Situation 1**. : _Only bearings can be transmitted among agents or obtained by detecting, then we can obtain \(\mathbf{X}_{ij}^{t}\) with Equations (37) and (38) if \(\text{rank}\big{(}\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\big{)}=N-N_{l}\) holds, where the number of positive elements must be greater than \(n\) for each row of \(\mathbf{\mathcal{L}}+\mathbf{\mathcal{L}}_{s}\)._  
**Situation 2**.