# Terminal Distributed Cooperative Guidance Law for Multiple UAVs Based on Consistency Theory
## 3 Design of Bottom Guidance Law Based on Sliding Mode Control Theory
### Time-to-Go Estimation of PNG Law  
Assuming that the UAV is guided by the PNG law, the acceleration is expressed as follows:  
\[a_{M}=NV_{M}\dot{\lambda} \tag{5}\]  
where, \(N\) is the navigation gain and \(\dot{\lambda}\) is the rate of the LOS angle.  
Substituting Equation (5) into Equation (3), yields  
\[\dot{\gamma}_{M}=N\dot{\lambda} \tag{6}\]  
Differentiating Equation (4) and substituting Equation (6), yields:  
\[\dot{\sigma}_{M}=\dot{\gamma}_{M}-\dot{\lambda}=(N-1)\dot{\lambda} \tag{7}\]  
Substituting Equation (2) into Equation (7), yields:  
\[\dot{\sigma}_{M}=-\frac{(N-1)V_{M}\sin\sigma_{M}}{r} \tag{8}\]  
It can be obtained from Equations (1) and (8) that:  
\[\frac{d\sigma_{M}}{dr}=\frac{\dot{\sigma}_{M}}{\dot{r}}=\frac{(N-1)\tan\sigma _{M}}{r} \tag{9}\]  
Figure 1: Relative motion relationship between unmanned aerial vehicle (UAV) and target.  
Integrating Equation (9) and its solution can be obtained as follows:  
\[r=r_{0}\left(\frac{\sin\sigma_{M}}{\sin\sigma_{M0}}\right)^{\frac{1}{N-1}} \tag{10}\]  
where \(r_{0}\) is the initial relative distance and \(\sigma_{M0}\) is the initial leading angle.  
Substituting Equation (10) into Equation (8), yields  
\[\dot{\sigma}_{M}=-(N-1)V_{M}\sin\sigma_{M}/r_{0}\left(\frac{\sin\sigma_{M}}{ \sin\sigma_{M0}}\right)^{\frac{1}{N-1}}=K(\sin\sigma_{M})^{\frac{N-2}{N-1}} \tag{11}\]  
where \(K=-\frac{(N-1)V_{M}}{r_{0}}(\sin\sigma_{M0})^{\frac{1}{N-1}}\).  
It can be obtained from Equation (11) that:  
\[dt=\frac{1}{K}(\sin\sigma_{M})^{\frac{2-N}{N-1}}d\sigma_{M} \tag{12}\]  
Integrating Equation (12) and using Taylor series expansion, ignore advanced items, \(\sin x=x-\frac{1}{6}x^{3}\) and \((1+x)^{\alpha}=1+\alpha x\), yields:  
\[\begin{array}{ll}t-t_{0}&=\frac{1}{K}\int_{\sigma_{M0}}^{\sigma_{M}}(\sin \sigma_{M})^{\frac{2-N}{N-1}}d\sigma_{M}\\ &\approx\frac{1}{K}\int_{\sigma_{M0}}^{\sigma_{M}}\left(\sigma_{M}-\frac{ \sigma_{M}^{2}}{6}\right)^{\frac{2-N}{N-1}}d\sigma_{M}\\ &=\frac{1}{K}\int_{\sigma_{M0}}^{\sigma_{M}}\sigma_{M}^{\frac{2-N}{N}}\left(1 -\frac{\sigma_{M}^{2}}{6}\right)^{\frac{2-N}{N-1}}d\sigma_{M}\\ &\approx\frac{1}{K}\int_{\sigma_{M0}}^{\sigma_{M}}\sigma_{M}^{\frac{2-N}{N-1 }}\left(1-\frac{2-N}{N-1}\frac{\sigma_{M}^{2}}{6}\right)d\sigma_{M}\\ &=\frac{1}{K}\int_{\sigma_{M0}}^{\sigma_{M}}\left(\sigma_{M}^{\frac{2-N}{N-1 }}+\frac{2-N}{N-1}\frac{\sigma_{M}^{\frac{N-1}{N}}}{6}\right)d\sigma_{M}\\ \end{array} \tag{13}\]  
Equation (13) can be further simplified as follows:  
\[t=t_{0}+\frac{r_{0}}{V_{M}}\left(1+\frac{2-N}{6(N-1)(2N-1)}\sigma_{M0}^{2} \right)\left(\frac{\sigma_{M0}}{\sin\sigma_{M0}}\right)^{\frac{1}{N-1}}-\frac {r_{0}}{V_{M}}\left(1+\frac{2-N}{6(N-1)(2N-1)}\sigma_{M}^{2}\right)\left(\frac {\sigma_{M}}{\sin\sigma_{M0}}\right)^{\frac{1}{N-1}} \tag{14}\]  
When the UAV attacks the target, the leading angle \(\sigma_{M}\) equals zero. Therefore, the time-to-go \(t_{go}\) at the moment \(t\) can be expressed as follows:  
\[t_{go}=\frac{r}{V_{M}}\left(1+\frac{2-N}{6(N-1)(2N-1)}\sigma_{M}^{2}\right) \left(\frac{\sigma_{M}}{\sin\sigma_{M}}\right)^{\frac{1}{N-1}} \tag{15}\]  
Defining,  
\[Nt=\frac{2-N}{6(N-1)(2N-1)} \tag{16}\]  
and the new time-to-go estimation method proposed in Equation (15) can be rewritten as follows:  
\[t_{go}=\frac{r}{V_{M}}\left(1+N\sigma_{M}^{2}\right)\left(\frac{\sigma_{M}}{ \sin\sigma_{M}}\right)^{\frac{1}{N-1}} \tag{17}\]  
Here, another time-to-go estimation method proposed in [32] is also given as below:  
\[t_{go}=\frac{r}{V_{M}}\left(1+\frac{\sigma_{M}^{2}}{2(2N-1)}\right) \tag{18}\]