# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 2 Point-Mass Model of UAV Dynamics
### Synthesis of UAV Adaptation Law
It was shown that sub-6 GHz mainstream MIMO can successfully deal with cellular selection and interference problems; in addition, the prospects for next-generation UAV communications were outlined and an assessment of the prospective benefits for UAVs was provided. In [PERSON] et al. [72], the authors discussed the main technological obstacles that stand in the way of further UAV communications development._  
Unlike [5; 23; 27; 73; 74; 75; 42], in the present paper we assume that the reference trajectory (15) along which the UAV group should move is given ahead of time. The group formation is set by the vector relative position (the offset vector) for each UAV with respect to trajectory (15), as follows:  
\[\delta_{i}=[\delta_{xi},\;\delta_{yi},\;\delta_{hi}]^{\rm T},\quad i=1,2, \ldots,N. \tag{49}\]  
Then, the reference trajectory for the \(i\)-th UAV \(i\) is provided by the following vector function:  
\[r_{i}(t)=[r_{x}(t)+\delta_{xi},\;r_{y}(t)+\delta_{yi},\;r_{h}(t)+\delta_{hi}]^ {\rm T}=[r_{xi}(t),\;r_{yi}(t),\;r_{hi}(t)]^{\rm T}. \tag{50}\]  
By choosing appropriate offsets \(\delta_{i}\), we can set the required formation of the UAV group. By analogy with (16) and (17), we can define the following variables for system (48):  
\[ \begin{array}{ll}e_{x1i}=x_{1i}-r_{xi}(t),&e_{x2i}=x_{2i}-\dot{r}_{xi}(t),\\ e_{y1i}=y_{1i}-r_{yi}(t),&e_{y2i}=y_{2i}-\dot{r}_{yi}(t),\\ e_{h1i}=h_{1i}-r_{hi}(t),&e_{h2i}=h_{2i}-\dot{r}_{hi}(t),\end{array}  \tag{51}\]  
where \(x_{1i}=x_{i}\), \(x_{2i}=\dot{x}_{i}\), \(y_{1i}=y_{i}\), \(y_{2i}=\dot{y}_{i}\), \(h_{1i}=h_{i}\), \(h_{2i}=\dot{h}_{i}\), and  
\[ \begin{array}{l}y_{xi}=\alpha_{xi}e_{x1i}+e_{x2i},\\ y_{yi}=\alpha_{yi}e_{y1i}+e_{y2i},\\ y_{hi}=\alpha_{hi}e_{h1i}+e_{h2i},\end{array}  \tag{52}\]  
where \(\alpha_{xi}>0\), \(\alpha_{yi}>0\), \(\alpha_{hi}>0\), \(i=1,2,\ldots,N\). With these variables, system (48) can be written as  
\[\dot{\varepsilon}_{x1i} =-\alpha_{xi}\varepsilon_{x1i}+y_{xi},\] \[\dot{\varepsilon}_{y1i} =-\alpha_{yi}\varepsilon_{y1i}+y_{yi}, \tag{53}\] \[\dot{\varepsilon}_{h1i} =-\alpha_{hi}\varepsilon_{h1i}+y_{hi},\]  
\[\dot{y}_{xi} =\alpha_{xi}y_{xi}-\alpha_{xi}^{2}\varepsilon_{x1i}+\frac{1}{m_{i} }\varepsilon_{1i}-\bar{r}_{xi}(t),\] \[\dot{y}_{yi} =\alpha_{yi}y_{yi}-\alpha_{yi}^{2}\varepsilon_{y1i}+\frac{1}{m_{i }}\bar{\varrho}_{2i}-\bar{r}_{yi}(t), \tag{54}\] \[\dot{y}_{hi} =\alpha_{hi}y_{hi}-\alpha_{hi}^{2}\varepsilon_{h1i}+\frac{1}{m_{i }}\sigma_{3i}-\bar{r}_{hi}(t)-g.\]  
Let the control goal for the \(i\)-th UAV be \(\lim\limits_{t\to\infty}Q_{ii}(e_{\text{xyhi}},y_{\text{xyhi}})\to 0\), where \(e_{\text{xyhi}}=(e_{x1i},e_{y1i},e_{h1i})\), \(y_{xyhi}=(y_{xi},y_{yi},y_{hi})\) are introduced. Then,  
\[Q_{ii}(e_{\text{xyhi}},y_{\text{xyhi}})=\frac{1}{2}\varepsilon_{x1i}^{2}+ \frac{1}{2}\varepsilon_{y1i}^{2}+\frac{1}{2}\varepsilon_{h1i}^{2}+\frac{m_{i }}{2}y_{xi}^{2}+\frac{m_{i}}{2}y_{yi}^{2}+\frac{m_{i}}{2}y_{hi}^{2}. \tag{55}\]  
Because no interaction between UAVs in a group is assumed, goal (55) can be achieved using the earlier described control loop (23) as follows:  
\[ \begin{split}\varrho_{1i}&=-\hat{m}_{i}(k_{xi}y_{xi }-\bar{r}_{xi}(t))+\hat{k}_{xi}y_{xi},\\ \varrho_{2i}&=-\hat{m}_{i}\big{(}k_{yi}y_{yi}-\bar{r }_{yi}(t)\big{)}+\hat{k}_{yi}y_{yi},\\ \sigma_{3i}&=-\hat{m}_{i}(k_{hi}y_{hi}-\bar{r}_{hi}(t )-g)+\hat{k}_{hi}y_{hi},\end{split}  \tag{56}\]  
where \(\hat{m}_{i}\), \(\hat{k}_{xi}\), \(\hat{k}_{yi}\), \(\hat{k}_{hi}\) are tunable controller parameters and \(k_{xi}>0\), \(k_{yi}>0\), \(k_{hi}>0\) stand for the fixed (design) parameters. The adaptation algorithm is as follows:  
\[ \begin{split}\hat{m}_{i}&=-\gamma_{mi}\Big{(}-k_{ xi}y_{xi}^{2}+y_{xi}\bar{r}_{xi}(t)-k_{yi}y_{yi}^{2}+y_{yi}\bar{r}_{yi}(t)-k_{hi}y_{hi} ^{2}+y_{hi}(\bar{r}_{hi}(t)+g)\Big{)},\\ \dot{k}_{xi}&=-\gamma_{xi}y_{xi}^{2},\\ \dot{k}_{yi}&=-\gamma_{yi}y_{yi}^{2},\\ \dot{k}_{hi}&=-\gamma_{hi}y_{hi}^{2},\end{split}  \tag{57}\]  
where \(\gamma_{mi}>0\), \(\gamma_{xi}>0\), \(\gamma_{yi}>0\), \(\gamma_{hi}>0\) are the adaptation gains (design parameters).