# Nonsingular Terminal Sliding Mode Based Finite-Time Dynamic Surface Control for a Quadrotor UAV
## 4 Simulation and Analysis  
### Numerical Simulation Results  
In this section, numerical simulations are applied to validate the effectiveness of the presented control algorithm. The quadrotor UAV model parameters are described as \(m=1.1\,\mathrm{kg}\), \(k_{i}(i=1,\,2,\,3)=0.01\,\mathrm{Ns/m}\), \(J_{i}(i=1,\,2)=1.22\,\mathrm{kg}\cdot\mathrm{m}^{2}\), \(g=9.81\,\mathrm{m/s}^{2}\), \(l=1\,\mathrm{m}\), \(c=1.5\), \(k_{i}(i=4,\,5,\,6)=0.012\,\mathrm{Ns/m}\) and \(J_{3}=2.2\,\mathrm{kg}\cdot\mathrm{m}^{2}\). It is assumed that the external disturbances and parameter uncertainties are respectively given as \(\boldsymbol{d}=\big{[}\ 0.1\cos(0.2t)\mathrm{N},\,0.5\mathrm{sin}(0.1t) \mathrm{N}.\mathrm{m},\,0.6\mathrm{cos}(0.5t)\mathrm{N}.\mathrm{m},\,0.2 \mathrm{sin}(0.2t)\mathrm{N}.\mathrm{m}\ \big{]}^{\mathrm{T}}\), and \(\Delta(\boldsymbol{I})=\big{[}\ 0.1\mathrm{cos}(0.2t)\mathrm{kg}\cdot\mathrm{m}^{2}, \,0.2\mathrm{sin}(0.3t)\mathrm{kg}\cdot\mathrm{m}^{2},\,0.2\mathrm{cos}(0.5t )\mathrm{kg}\cdot\mathrm{m}^{2}\ \big{]}^{\mathrm{T}}\).  
The parameters of ESO presented in (20) are \(\zeta=0.0001,\,\rho=0.0001,\,r_{1}=2,\,r_{2}=0.02\), \(b_{1}=0.9\), \(L_{1}=1\), \(L_{2}=0.2\). The gains of position control law are chosen as \(l_{1}=1\), \(c_{1}=0.8\), \(\lambda_{1}=0.2\), \(\lambda_{2}=0.3\), \(p_{1}=0.6\), \(p_{2}=0.2\), \(\varepsilon=0.5\), \(m_{1}=0.1\), and \(n_{1}=0.1\). The main parameters of the attitude subsystem are \(r_{31}=2\), \(r_{32}=1\), \(r_{33}=0.6\), \(r_{41}=0.05\), \(r_{42}=0.5\), \(r_{43}=0.2\), \(b_{3}=0.8\), \(\varepsilon_{21}=2\), \(\varepsilon_{22}=5\), \(\varepsilon_{23}=1\), \(c_{21}=0.9\), \(c_{22}=0.9\), and \(c_{23}=0.8\).  
To illustrate the superiority of the proposed control scheme, a finite-time dynamic surface control (FTDSC) scheme [14] is introduced. External disturbances and parameter uncertainties are restricted to the same values in the proposed algorithm and FTDSC to make a fair comparison. Figure 2 shows a comparison of trajectory/attitude tracking under the proposed scheme and the FTDSC scheme. It is observed that the proposed scheme obtains a better performance in terms of tracking the desired trajectory/attitude. Figure 3 compares the tracking errors. It is shown that the proposed scheme has better convergence performance. A comparison of the linear/angular velocity is shown in Figure 4. It can be concluded that the proposed scheme provides better stability. Figure 5 illustrates the control inputs under the proposed scheme and the FTDSC scheme. It can be clearly seen that the control inputs of the proposed scheme are appropriate. Figure 6 describes the convergence performance of the observation errors of the ESOs. It is obvious that the proposed ESOs can estimate the actual disturbances successfully with a settling time of less than 10 s. Therefore, highly precise tracking control can be accomplished via the proposed scheme.  
The root-mean-square error (RMSE) and the mean absolute error (MAE) were used as performance indicators to assess the results of the comparison simulation. The results are listed in Table 1. Overall, the proposed scheme performs better than the FTDSC. Although the RMSE of the tracking errors of \(\phi\) and \(\theta\) in the proposed scheme are larger than that of FTDSC, the tracking errors of \(\phi\) and \(\theta\) in the proposed scheme have better stability.  
Figure 3: Time response of trajectory/attitude tracking errors.  
Figure 2: Time response of trajectory/attitude tracking.  
Figure 4: Time response of linear/angular velocity.  
Figure 5: Time response of control inputs.