# Speed-Gradient Adaptive Control for Parametrically Uncertain UAVs in Formation
## 5 Simulation Results
### UAV Group Control under External Disturbances  
To test the robustness properties of the obtained algorithms and compare them with other algorithms that solve similar problems, the results by [PERSON] et al. [26] for a four-agent formation were simulated. Control laws for group control (61) and (62) and tuning algorithms (69) were used.  
The desired trajectory was  
\[r_{x}(t)=60t\,\mathrm{m},\quad r_{y}(t),\equiv 0,\quad r_{h}(t)=80\,\mathrm{m}. \tag{72}\]  
Figure 9: Thrust forces for the case of a group of variable mass UAVs control with initial mass values of 500 kg (**a**), 450 kg (**b**), 400 kg (**c**), and 350 kg (**d**).  
The following nominal parameters of the UAV were used: mass \(m=20\) kg; wing area \(S=1.37\) m\({}^{2}\); air density \(\rho_{a}=1.225\) kg/m\({}^{3}\) ; zero-lift drag coefficient \(C_{D_{0}}=0.02\); induced drag factor \(k_{c}=0.1\). In the simulation, at the 25-th second of the flight the coefficients \(C_{D_{0}}\) and \(k_{c}=0.15\) increased abruptly by 50% (up to \(C_{D_{0}}=0.03\), \(k_{c}=0.15\)).  
The formation was configured by offsets (49) and (50) relative to the reference trajectory  
\[\delta_{x}=[0,-100,0,100]^{\mathrm{T}},\quad\delta_{y}=[100,0,-100,0]^{\mathrm{ T}},\quad\delta_{h}=[0,0,0,0]^{\mathrm{T}}. \tag{72}\]  
The control laws (52), (61), (62) and tuning algorithm (69) parameters were identical for all the UAVs in the formation. In (52), the following output parameters \(y_{xi}\), \(y_{yi}\), \(y_{hi}\) were used: \(\alpha_{xi}=\alpha_{yi}=\alpha_{hi}=0.2182\). The constant parameters of control law (61) were set to \(k_{xi}=k_{yi}=k_{hi}=0.4583\). The following adaptation algorithm (47) parameters were used: \(\gamma_{mi}=15.5\), \(\sigma_{mi}=0.008\), \(\gamma_{xi}=\gamma_{yi}=\gamma_{hi}=15.5\), \(\sigma_{xi}=\sigma_{yi}=\sigma_{hi}=0.008\), and the initial values of adjustable parameters were set to \(m_{0i}=20\) kg, \(\hat{k}_{x0i}=\hat{k}_{y0i}=\hat{k}_{h0i}=-100\).  
The consensus algorithm in form (62) was used to implement the group control. Data exchange between agents was carried out according to the connections graph shown in Figure 1. The consensus algorithm parameters were as follows: \(k_{xc}=k_{yc}=k_{hc}=850\), \(\rho_{x}=\rho_{y}=\rho_{h}=700\). The initial parameters were the same as in [PERSON] et al. [26].  
The following common initial conditions were used for all UAVs in the group: initial attitudes \(\gamma(0)=0\) rad, \(\psi(0)=0\) rad.  
For the first UAV, the initial uncertain mass was \(m=24\) kg with initial coordinates \(x(0)=0\) m, \(y(0)=200\) m, \(h(0)=95\) m, speed \(V_{g}(0)=70\) m/s. For the second UAV, the initial uncertain mass was \(m=17\) kg with initial coordinates \(x(0)=0\) m, \(y(0)=60\) m, \(h(0)=90\) m, speed \(V_{g}(0)=60\) m/s. For the third UAV, the initial uncertain mass was \(m=22\) kg with initial coordinates \(x(0)=0\) m, \(y(0)=-200\) m, \(h(0)=70\) m, speed \(V_{g}(0)=40\) m/s. For the fourth UAV, the initial uncertain mass was \(m=22\) kg with initial coordinates \(x(0)=0\) m, \(y(0)=-60\) m, \(h(0)=80\) m, speed \(V_{g}(0)=50\) m/s.  
In addition, in the simulations the masses of agents changed linearly with the identical slope of \(-0.083\) kg/s.  
External perturbations \(d_{x}\), \(d_{y}\), \(d_{h}\) were added to the first three equations of (1). The external disturbances were the same as in [PERSON] et al. [26], namely, \(d_{x1}=0.5\cos(t)\), \(d_{y1}=0.8\sin(t)\), \(d_{h1}=0.5\cos(t)\), \(d_{x2}=4\sin(t)\), \(d_{y2}=3\sin(t)\), \(d_{h2}=4\sin(t)\), \(d_{x3}=cos(2t)\), \(d_  
Figure 11: Tracking error comparison curves of three positions and three velocities.  
Figure 10: 3D tracking trajectories.