# Quadcopter Adaptive Trajectory Tracking Control:
## 4 Numerical Results  
In order to verify the proposed method in this paper, we evaluate the simulation results of the Unmanned Aerial Vehicle-Quadcopter following several desired tracking trajectories. The parameters of the proposed system are given as: \(m=1.568\) kg, \(I_{xx}=I_{yy}=0.0119\) kg\(\cdot\)m\({}^{2}\), \(l_{zz}=0.0223\) kg\(\cdot\)m\({}^{2}\), \(g=9.8\) m/s\({}^{2}\), \(b=7.73\times 10^{-6}\) Ns\({}^{2}\), \(d=1.28\times 10^{-7}\) Nms\({}^{2}\), \(\mu_{1}=\mu_{2}=\mu_{3}=0.1\) s\({}^{-1}\) and the initial states are chosen such that an initial position of the quadcopter lies outside the desired trajectory to strongly verify the proposed method: \(x(0)=0.5\) m, \(y(0)=0.5\) m, \(z(0)=0\) m, \(\phi(0)=\theta(0)=\psi(0)=0\) rad, \(x_{d}(0)=0\) m, \(y_{d}(0)=0\) m, \(z_{d}(0)=0\) m, \(\hat{\Gamma}(0)=[1,1,1,1]^{T}\), \(\text{vec}\big{(}\hat{E}\big{)}=[0.5,0.5,0.5,0.5]^{T}\). Based on the proposed Theorem 1, the matrix and scalar coefficients are selected as follows: \(\mathbf{K}_{1}=\mathbf{I}_{2}\), \(\mathbf{K}_{2}=2\mathbf{I}_{2}\), \(\mathbf{K}_{3}=3\mathbf{I}_{2}\), \(\mathbf{K}_{4}=16\mathbf{I}_{2}\), \(k_{5}=6\), \(k_{6}=5\), \(\epsilon=0.2\), \(\mathbf{Q}_{1}=10\mathbf{I}_{4}\), \(\mathbf{Q}_{2}=4\mathbf{I}_{4}\). The simulation scenarios involve several flying shapes in the air from where the initial position does not belong to these desired paths. The objective is that the investigated system should be able to track given trajectories using finite time simulation and satisfying Assumption 2. To generate several complicated orbits, the desired \(\eta\) polygon horizontal trajectory is created in two-dimensional space [27] as follows:  
\[ \begin{split} x_{d}&=a\left[R+\cos\left(\eta\omega t \right)\right]\cos\left(\omega t\right)\\ y_{d}&=a\left[R+\cos\left(\eta\omega t\right) \right]\sin\left(\omega t\right)\\ z_{d}&=\left[h_{d}-z(0)\right]\left(1-e^{-\omega t }\right)+z(0)\end{split}  \tag{58}\]  
where \(a\), \(\omega\), and \(R\) are the scale, the angular velocity, and radius of the desired trajectory, respectively. They were set to \(a=2.5\), \(\omega=0.5\) rad/s, and \(R=1\) m. It can be observed that the reference trajectories can be generated by changing \(\eta\) in (58) to obtain \(\eta\)-sided smooth polygon. We either set \(\eta=6\) (Figure 2(a)) or \(\eta=10\) (Figure 2(b)) in \(xyz\)-space and \(xy\)-plane and desired altitude \(h_{d}=4\) m to generate two reference shapes to verify the proposed approach. Additionally, to emphasize the effectiveness of the proposed scheme, the adaptive sliding mode control [25] and Multi-loop PID [46] are also applied to this tracking problem. These methods show that the position and attitude subsystems are guaranteed to be in stability by two separated controllers, but there is not any relationship between these two loops. Therefore, we expect our method presented in the red box [1] in Figure 2(a) to be able to cope with issue. Figure 2: Control diagram of the proposed method. Figure 3(a),b compares the performance of three different approaches which include the proposed method, adaptive sliding mode control, and Multi-loop PID. Having started from the same initial state, the 3D position tracking errors tend towards the origin. If we increase the model parameters by 30% to investigate the system robustness, only the blue solid line asymptotically converges to the origin because of the proposed adaptive law. The estimated values of \(\Gamma,\mathbf{E}\) in Figure 4(a),b become constant as time goes to infinity, which verifies our Theorem 1. The remaining curves exhibit oscillations in the tracking errors along the \(x\) and \(y\)-axes from the fifth to the twentieth second. Moreover, when it comes to considering the \(z\)-axis, the error of altitude tracking asymptotically converges to the origin and is always limited by \(\epsilon\) in (46) to ensure Assumption 2. Figure 3: Trajectory tracking performance of the proposed method in the \(xyz\)-space and \(xy\)-plane.