# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 2 Point-Mass Model of UAV Dynamics
### Synthesis of UAV Adaptation Law
\(\square\)  
**Remark 1**. : _Relations (17) can be treated as an implicit reference model [29; 30; 53; 56], where the parameters \(\alpha_{x}\), \(\alpha_{y}\), \(\alpha_{h}\) set the goal function (18) convergence rate._  
**Remark 2**. : _The efficiency of the adaptation algorithm (45) is preserved if the mass \(m\) in (11) is time-varying with a sufficiently small rate of change. The rigorous consideration of the case of the time-varying parameters can be performed based on the recent results in [57; 58]._  
**Remark 3**. : _The adaptation algorithm (45) is a combined one. It includes the identification procedure provided by the first expression in (45) and the direct adaptive control described by the last three expressions in (45)._  
**Remark 4**. : _In practice, it is advisable to use the following regularized form of the adaptation algorithm (45), cf. [53]:_  
\[ \begin{split}\hat{m}&=-\gamma_{m}\Big{(}-k_{x}y_{x} ^{2}+y_{x}\bar{r}_{x}(t)-k_{y}y_{y}^{2}+y_{y}\bar{r}_{y}(t)-k_{h}y_{h}^{2}+y_{ h}(\bar{r}_{h}(t)+g)\Big{)}-\sigma_{m}(\hat{m}-m_{0}),\\ \hat{k}_{x}&=-\gamma_{x}y_{x}^{2}-\sigma_{x}\Big{(} \hat{k}_{x}-\hat{k}_{x0}\Big{)},\\ \hat{k}_{y}&=-\gamma_{y}y_{y}^{2}-\sigma_{y}\Big{(} \hat{k}_{y}-\hat{k}_{y0}\Big{)},\\ \hat{k}_{h}&=-\gamma_{h}y_{h}^{2}-\sigma_{h}\Big{(} \hat{k}_{h}-\hat{k}_{h0}\Big{)},\\ \end{split}  \tag{47}\]  
_where \(\sigma_{m}>0\), \(\sigma_{x}>0\), \(\sigma_{y}>0\), \(\sigma_{h}>0\) are parametric feedback gains (design parameters) and \(m_{0}\), \(\hat{k}_{x0}\), \(\hat{k}_{y0}\), \(\hat{k}_{h0}\) are \"guessed\" values of the control law (23) tunable coefficients chosen based on the prior information of the UAV parameters._  
**4. Control of Multi-Agent UAV Formation**  
_4.1. Problem Description_  
Consider a group of \(N\) UAVs (agents) indexed as \(i=1,2,\ldots,N\). The dynamics of each UAV can be described by (11), as follows:  
\[ \begin{split}\dot{x}_{i}&=\frac{1}{m_{i}}\bar{ \sigma}_{1i},\\ \dot{y}_{i}&=\frac{1}{m_{i}}\bar{\sigma}_{2i},\\ \ddot{\bar{u}}_{i}&=\frac{1}{m_{i}}\dot{u}\bar{v}_{ 3i}-g,\end{split}  \tag{48}\]where \(m_{i}\) is the mass of the \(i\)-th UAV and \(\vartheta_{1i}\), \(\vartheta_{2i}\), \(\vartheta_{3i}\) are the virtual controls for \(i\)-th UAV. Here, masses \(m_{i}\) are assumed to be unknown within the known range \(m_{\min i}\leqslant m_{i}\leqslant m_{\max i}\), \(i=1,2,\ldots,N\). **Remark 5**. : _UAV communication channels and data exchange protocols are not considered in the present paper, and it is assumed that the information exchange necessary to implement the proposed algorithms is feasible in one form or another. Results regarding data transfer between UAVs in formation can be found in [23; 37; 59; 60; 61; 62; 63; 64; 65; 66; 67]. Particularly, in [59; 63; 66; 67], adaptive coding procedures for navigation data exchanged between the UAVs in formation was studied in depth. For satellite communications, [PERSON] et al. [69] solved the problem of safe and efficient beamforming in multi-agent satellite systems to simultaneously achieve data transmission security against interception and low power consumption. [PERSON] et al. [70] explored multicast satellite and aerial-integrated network with rate-splitting multiple access, where the satellite and UAV components are controlled by the network control center and operate in the same frequency band. An optimization problem was formulated to maximize the total speed in terms of the signal-interference ratio and power limitations per antenna in the UAV and satellite. [PERSON] et al. [71] addressed the aim of providing efficient satellite-to-satellite communication while minimizing the overall transmitted signal power and meeting the bit-rate requirements for multiple blocked users. In [72], a broad overview of the state-of-the-art of UAV communications from an industrial point of view was provided.