## 6 Simulation Results  
In this section, we consider a simulation scenario that four UAVs track the target vehicle in the predefined formation. The simulation is carried out on a personal computer, which has 6 CPU cores (Intel(R) Core(TM) i7-10750H CPU @ 2.60 GHz), and the simulation environment is Matlab2020a. _fitrap_ function is used for Gaussian process regression.  
The unknown parts of dynamic function are given as follows  
\[\begin{array}{l}f_{i1}(v_{i},\theta_{i})=-0.0035|v_{i}-13|\exp(\sqrt{0.8v_{i }}),\\ f_{i2}(v_{i},\theta_{i})=-0.04\sqrt{v_{i}}\sin 3\theta_{i}+\frac{\ln v_{i}}{v_{i }^{1.5}}\cos 2\theta_{i},\end{array} \tag{29}\]  
where \(i=1,2,3,4\). The 2500 training inputs are equally distributed on the set \([8,20]\times[-\pi,\pi]\), since the velocity of the UAV is subject to some constraints. The labels in the training pairs are the observed value of Equation (29) with Gaussian noise \(\varsigma\), \(\varsigma\sim\mathcal{N}(0,0.03)\).  
The parts of disturbance, \(d_{i1}\) and \(d_{i2}\), are given as the white noise and bounded noise, respectively, which account for 2% of control input. The trajectory of the target vehicle is given as follows  
\[\begin{array}{l}x_{0}(t)=200\cos 0.07t,\\ y_{0}(t)=150\sin 0.07t,\end{array} \tag{30}\]  
where \((x_{0},y_{0})\) is the position of the target vehicle.  
Figure 3 shows interactions among the UAVs and the target vehicle, which is chosen under Assumption 1. The control gains are chosen to be \(k_{1}=1.5\), \(k_{2}=0.5\), the diagonal matrix \(\mathbf{B}=diag\{1,1,0,0\}\), the adjacency matrix \(\mathbf{A}\) and matrix \(\tilde{L}\) are given as follows  
\[\mathbf{A}=\begin{bmatrix}0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\end{bmatrix},\qquad\qquad\tilde{\mathbf{L}}=\begin{bmatrix}3&-1&0&-1\\ -1&3&-1&0\\ 0&-1&2&1\\ -1&0&-1&2\end{bmatrix}.\]  
The whole simulation lasts for 200 seconds, during the simulation, the estimation error of GP model is bounded, which can be seen in Figure 4. It is noting that Gaussian noise \(\varsigma\) is of great importance in the data learning of GP model, since too much noise would reduce the reliability of the training data.  
To visually demonstrate the tracking performance, Figure 5 shows the trajectory of four UAVs and the target in the three-dimensional plots, where all UAVs track the target vehicle in the predefined formation. Figure 6 illustrates the consensus tracking error of each UAV, which is defined in Equation (19). It is noting that the consensus tracking error of each UAV converges to a certain bound exponentially and the final bounds of error in two directions are 1 m.  
Figure 4: Estimation error of GP model.  
Figure 3: The communication graph of the vehicles.  
To show the effectiveness of the proposed method, we consider the other three cases, which include the case based on NNs model, based on polynomial model and based on nominal model. We employ a two-layer feed-forward network via _Neural Net Fitting APP_ in Matlab to train NNs model of unknown dynamic, and the number of hidden neurons is 5 in the training. The polynomial model is trained via _Curve Fitting APP_ in Matlab, and the degree of the polynomial is 3. Table 1 demonstrates the RMSE of three models, among which the RMSE of GP model is minimal. Figure 7 shows the accumulated consensus tracking errors of 4 UAVs, which are defined as \(\sum_{i=1}^{4}\epsilon_{i,j}\), where \(j=1,2\) denotes two directions. We can find that the tracking performance based on GP model and NNs model are almost the same, which are better than that based on polynomial model. The performance based on nominal model, without considering uncertainty, is the worst. Although the performance based on GP model and NNs model are almost the same, the uniform error of GP model can be explained based on Lemma 3.  
Figure 5: Trajectories of 4 UAVs and Target vehicle.  
Figure 6: Consensus tracking error of four UAVs.