## 2 Theoretical Basis of Time-Varying Target Formation
### Multi-Agent System Control Law Design Based on Graph Theory and Consistency Algorithm  
The corresponding control law for the second-order integrator model is designed to solve the multi-agent affine maneuver formation in this section. As a consequence, consider a network composed of \(N\) agents in \(R^{d}\) space, given \(d\geq 2\) and \(N\geq d+2\) in this system. For the switching topology, in the time period of \(t\rightarrow\infty\), there are countless time intervals \([t_{m},t_{m+1}),m\in Z\),where \(t_{0}=0\) represents time zero. Assume that each time interval satisfies \(t_{m+1}-t_{m}\geq T\), where \(T>0\). At every time interval, there are \(k+1\) non-overlapping sub-intervals \([t_{m},t_{m+1}),s=0,1,\cdots,k\). It can be assumed that within each sub-interval \([t_{m},t_{m+1})\),the network topology \(G_{s}\) (switching topology) is fixed until the next time sub-interval is entered. The corresponding stress matrix can be expressed as \(\Omega_{s},s=0,1,\cdots,k\)[24]. **Assumption 1** (Affine span of nominal formation): _For the nominal formation \((G_{s},r_{s})\), assume that \(\left\{r_{s}\right\}_{i=1}^{n}\) affinely span \(R^{d}\)._  
Figure 4: Target allocation in static case. (**a**) The seventh agent in the sequence fails, leaving six agents. (**b**) A random fourth agent fails, leaving six agents. (**c**) The sixth agent in the sequence fails, leaving five agents. (**d**) A random fifth agent fails, leaving five agents. Based on the switching topology networks, \(p_{si}\in R^{d},i=\{1,2,\cdots,N\}\) represents the position of agent \(i\) in the \(s\)-th switching topology, and the position of the entire formation are denoted as \(p_{s}=[p_{s1}^{T},p_{s2}^{T},\cdots,p_{sN}^{T}]^{T}\in R^{dN}\). Information is transferred between agents by switching topologies \(\mathsf{G}_{s}\). **Assumption 2** (Stress matrix of nominal formation). : _Assume that the nominal formation \((G_{s},r_{s})\) has a positive-semidefinite stress matrix \(\mathsf{\Omega}_{s}\) satisfying \(\text{rank}(\mathsf{\Omega}_{s})=n-d-1\)._  
If the set of all multi-agents is defined as \(\mu_{s}\), \(\mu_{sl}=[1,2,\cdots,N_{l}]\) express the leaders set and the follower set is \(\mu_{sf}=\mu_{s}/\mu_{sl}\).The formation of leader is \(p_{sl}=[p_{s1}^{T},p_{s2}^{T},\cdots,p_{sN_{l}}^{T}]^{T}\), and the formation of follower is \(p_{sf}=[p_{sN_{l+1}}^{T},p_{sN_{l+2}}^{T},\cdots,p_{sN}^{T}]^{T}\). Hence, the formation of the whole formation can be rewritten as \(p_{s}=[p_{sl}^{T},p_{sf}^{T}]^{T}\). In this network, assuming that there is no information transfer between leaders, and the leaders can't obtain the information of followers. From the-above mentioned analysis, the stress matrix can be divided as  
\[\mathsf{\Omega}_{s}= \begin{bmatrix}\mathsf{\Omega}_{s_{ll}}&\mathsf{\Omega}_{s _{lf}}\\ \mathsf{\Omega}_{s_{fl}}&\mathsf{\Omega}_{s_{ff}}\end{bmatrix} = \begin{bmatrix} 0&0\\ \mathsf{\Omega}_{s_{fl}}&\mathsf{\Omega}_{s_{ff}}\end{bmatrix}  \tag{13}\]  
**Definition 2** (Affine localizability). : _The nominal formation \((G_{s},r_{s})\) is affinely localizable by the leaders if for any \(p_{s}=[p_{sl}^{T},p_{sf}^{T}]^{T}\in A(r)\), \(p_{sf}\) can be uniquely determined by \(p_{sl}\)._  
The Definition 2 can be interpreted as if the formation is in \(A(r)\), then the position of followers can be uniquely determined by the leaders. As a result, this definition guarantees that the followers can track any desired configuration. **Assumption 3** (Affine localizability of nominal formation). : _Assume that the nominal formation \((G_{s},r_{s})\) is affinely localizable by the leaders._  
Before the control law is designed, Assumptions 1-3 need to be satisfied. To satisfy Assumption 1, the multi-agent system must be affinely span in \(R^{d}\). To satisfy Assumption 2, the nominal configuration needs to be universally rigid. For the purpose of Assumption 3, there are at least \(d+1\) agents selected as leaders that capable of affine span \(R^{d}\) in the nominal formation.