# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 7 Conclusions  
In this paper, we have proposed a new feedback-linearized UAV model with point mass. A linear model with an unknown value of the UAV mass was obtained via transformation. The velocity gradient method was applied to obtain a combined adaptive control law that includes equations for estimating the unknown mass value and for tuning the parameters of a linear controller with an implicit reference model. The obtained new adaptive algorithms were used to solve the problem of decentralized control of a UAV formation with an unknown mass based on a consensus algorithm. Rigorous stability conditions for control and identification are presented. Computer simulations of the obtained algorithms for both a single UAV and a UAV formation show good performance of the proposed approach under conditions of variable UAV mass. Further research might involve considering the influence of wind on the movement of the UAVs, as well as considering situations in which not every UAV has access to information about the movement trajectory.  
Conceptualization, A.M.P.; data curation, D.G.K. and A.A.S.; formal analysis, A.M.P.; funding acquisition, A.M.P.; investigation, D.G.K. and A.A.S.; methodology, A.M.P.; project administration, A.M.P.; software, A.M.P., D.G.K. and A.A.S.; supervision, A.M.P.; writing--original draft, A.M.P. and B.A.; writing--review and editing, A.M.P. and B.A. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{p{14.2 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**\#** & **Other Works** & **This Work** \\ \hline
1 & [5] – tracking the moving ground target in the real-time; method: fuzzy-logic model-reference adaptive control (FL MRAC) & – tracking the given reference trajectory; method: implicit reference model (IRM) adaptive control \\
2 & [20] – UAV parameters are known; method: non-adaptive control [27] – UAV type: VTOL; control vector: command forces and torques; UAV parameters are known; method: adaptive estimation of disturbance [26] – control vector: \(u=[F_{\text{thrust}},n_{\text{lf}},\varphi]^{\text{T}}\); uncertain parameters: \(C_{\text{Dv}},k_{c},S\); method: robust adaptive controller [28; 31; 32] – the reference trajectory is known only to the leading UAV; no consensus is ensured; UAV type: rotating-wing (quadrotor) [41] – single UAV; regulation of \(V_{g}\), \(\gamma\), tracking \(\psi^{*}(t)\); point-mass dynamics model without kinematic relations; control vector: \(u=[F_{\text{thrust}},n_{\text{lf}},\varphi]^{\text{T}}\); uncertain parameters: \(C_{\text{Dv}},k_{c}\); method: adaptive control for non-affine plants [42] – mission: passing a waypoint by several UAVs; control vector: \(u=[F_{\text{thrust}},n_{\text{lf}},\varphi]^{\text{T}}\); parameters are known; method: optimal design of the behavioral approach of decentralized control; communication graph: undirected (ring topology) [43] – tracking the desired trajectory on the plane \((x_{d}(t),y_{d}(t))\); model: kinematic on the plane at constant height; control vector: \(u=[V_{g},\varphi]^{\text{T}}\); parameters: estimated by the Gaussian Process Regression training procedure; – communication graph: undirected & – tracking the desired trajectory in 3-\(D\) space; model: 3-\(D\) nonlinear point model; control vector: \(u=[F_{\text{thrust}},C_{\text{L}},\varphi]^{\text{T}}\); uncertain parameters: \(m(t)\), \(C_{\text{Dv}},k_{c}\); communication graph: directed \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison with other works.  
**Funding:** This work was carried out in BSTU \"VOENMEH\" with financial support from the Ministry of Science and Higher Education of the Russian Federation (government contract agreement No. 075-03-2020-045/2 of 9 June 2020).  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.