# Quadrotor Stabilization and Tracking Using Nonlinear Surface Sliding Mode Control and Observer
## 6 Simulation Results  
In this section, to verify the effectiveness of the proposed controller design method, simulations using SMC and SMO-based NSMC methods in the quadrotor system were demonstrated. In our simulation, the control system of a quadrotor is modeled using MATLAB/Simulink with ODE 45 as a solver. The disturbance model of the simulation is supposed to follow white Gaussian noise. Table 1 shows quadrotor dynamic system parameters used in the simulation. Moreover, we set the control parameters \(\lambda_{1,\cdots,A}\), \(\lambda_{x,y}\), \(\mu_{1,\cdots,8}\), \(\mu_{9,10}\), \(\mu_{11}\), and \(\mu_{12}\) in the simulation as 2, 0.3, 30, 0.1, 0.08, and 0.05, respectively. First, we confirm the estimation performance to prove the improvement of the quadrotor control system by SMO. For the quadrotor attitude control, Figure 5 shows the stabilization of the quadrotor angle \((\phi,\theta)\), which is initially slanted by 17 degrees. In Figure 5, the dotted line is a stabilization result by conventional SMC and the solid line is a stabilization result by SMO-based nonlinear surface sliding mode control (NSMC). We can confirm that the performance of NSMC combined with SMO is better than that of conventional SMC. The settling time for the angle \((\phi,\theta)\) of a quadrotor is reduced with the use of SMO-based NSMC. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m\) & 0.23 & [kg] \\ \(l\) & 0.215 & [m] \\ \(l_{x}\) & \(5.136\times 10^{-3}\) & [Ns\({}^{2}\)/rad] \\ \(l_{y}\) & \(5.136\times 10^{-3}\) & [Ns\({}^{2}\)/rad] \\ \(l_{z}\) & \(1.0616\times 10^{-2}\) & [Ns\({}^{2}\)/rad] \\ \(l_{r}\) & 1.4193 & [Ns\({}^{2}\)/rad] \\ \(g\) & 9.8 & [m/s\({}^{2}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical parameters of quadrotor model. Figure 6 shows that the quadrotor reaches the target position \((7,7,5)\) m from the initial position \((3,3,0)\) m, following the reference trajectory. In Figure 6, the SMO-based control shows stable tracking performance following the reference trajectory and has a small perturbation around the turning points. Figure 7 shows the trajectory projected onto the \(x-y\) plane. To compare tracking performance, we simulate the trajectory shown in Figure 6 using the SMO-based NSMC and conventional SMC without an observer. The quadrotor tracking starts at the point \((3,3)\) m and reaches the final point \((7,7)\) m by passing through several turning points. The SMO-based NSMC shows robustness against the effects of measurement noise. However, the conventional SMC trajectory shows SMC without an observer. The quadrotor tracking starts at the point \((3,3)\) m and reaches the final point \((7,7)\) m by passing through several turning points. The SMO-based NSMC shows robustness against the effects of measurement noise. However, the conventional SMC trajectory shows chattering caused by the measurement noise. We can confirm that the trajectory driven by the SMO-based NSMC reaches the target point with less fluctuation. Figure 5: Stabilization of quadrotor with a slanted initial angle. Figure 6: Quadrotor tracking system from initial position to target position. Figure 8 shows the tracking performance of a quadrotor. The trajectory of the quadrotor with disturbances and the simulation results along the \(x\)-, \(y\)-, and \(z\)-axes are shown in this figure. While the conventional SMC shows low performance with slow convergence speed and large steady state error due to disturbance, the proposed NSMC shows more robust performance against disturbance since the proposed augmented dynamics with correction variables represents the quadrotor system precisely and NSMC reduces the delay of reaching time. This robustness against disturbance is possible owing to the state estimation precision of SMO. We can confirm the tracking performance on the \(x\)-, \(y\)-, and \(z\)-axes in Figure 8, wherein the nonlinear-sliding-surface-based SMC reduces bounces, settling time, and rising time more than conventional SMC.