# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 8 Numerical Example  
Let us investigate the stabilization of the lander by the example of an object with typical characteristics. Consider the model (3), (4) with the parameters indicated in Table 1 and the IC presented in Table 2. The nominal values of the characteristics of the lander, reaction engines, measuring instruments and the Earth's atmosphere (GOST 4401-81), as well as their typical spreads [31] are used (Table 3). The balancing position of the lander and the programmed value of the roll angle (for controlling the trajectory of the lander) are determined by the components of the vector  
\[\theta_{pr}^{E+Q_{o}}=\big{[}\gamma_{pr}\quad\beta_{bal}\quad\alpha_{bal} \big{]}^{T}=\big{[}\ \pm 60^{\circ}\quad 0\quad-21.5^{\circ}\ \big{]}^{T}.\]  
The position of the measuring base of the GAMS (\"frozen\" orbital CS at the moment of powering on the GAMS) relative to the reference base (\"frozen\" Greenwich CS) is set, respectively, by the angles of course, longitude and latitude  
\[\eta_{gyr}=15^{\circ},\quad\lambda_{gyr}=31^{\circ},\quad\varphi_{gyr}=35^{ \circ}.\]  
The initial moment of time, counted from the moment of powering on the GAMS, as well as the IC for the motion of the CM in the study of various processes of angular motion are taken to be the same. Combinations of the IC of angular motion (64 variants) are used in statistics when testing control algorithms. We will call the set of ICs characteristic,  
\[\gamma_{0}=\gamma_{pr}-5^{\circ},\quad\beta_{0}=\beta_{bal}-5^{\circ},\quad \alpha_{0}=\alpha_{bal}-5^{\circ},\quad\omega_{x_{v}0}=\omega_{y_{v}0}^{E_{v}}= \omega_{z_{v}0}^{E_{v}}=-2^{\circ}/\text{s} \tag{44}\]  
and further we will use it for a visual graphical comparison of control processes with the same ICs on the existing and new algorithms. MATLAB simulates 64 samples (Table 2) out of 100 lander stabilization processes without roll overturns according to the standard algorithm. The worst statistical characteristics of the processes for samples are presented in Table 4, and the graphs of the corresponding process under typical IC (44) are shown in Figure 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline total simulation time & \(T=280\) s \\ numerical integration method & Runge-Kutta 4 th Order Method \\ size and board steps & \(h_{m}=0.005,\quad h_{b}=0.020\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline the initial moment of time & \(t_{0}=280\) s \\ (from the time of powering on the GAMS) & \\ height above the surface of the earth & \(h_{0}=80\) km \\ longitude (east) & \(h_{m}=0.005,\quad h_{b}=0.020\) \\ longitude (east) & \(\lambda_{0}=51^{\circ}\) \\ latitude (north) & \(\varphi_{0}=38^{\circ}\) \\ the angle of inclination of the trajectory & \(\theta_{0}=-1.5^{\circ}\) \\ linear velocity of the CM of the lander (relative & \(v_{0}=7.5\) km/s \\ to the Earth) & \(v_{0}=7.5\) km/s \\ heading angle & \(\eta_{0}=0\) \\ speed roll angle & \(\gamma_{0}=\gamma_{pr}\pm 5^{\circ}\) \\ angles of side slip & \(\beta_{0}=\beta_{bal}\pm 5^{\circ}\) \\ angles of attack & \(\alpha_{0}=a_{bal}\pm 5^{\circ}\) \\ angular velocity of channel \(x_{v}\) & \(\omega_{x_{v}0}^{E_{v}}=\pm 2^{\circ}/\text{s}\) \\ angular velocity of channel \(y_{v}\) & \(\omega_{x_{v}0}^{E_{v}}=\pm 2^{\circ}/\text{s}\) \\ angular velocity of channel \(z_{v}\) & \(\omega_{z_{v}0}^{E_{v}}=\pm 2^{\circ}/\text{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial Simulation Conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel** & **Moment of Inertia, kg-m\({}^{2}\)** & **Engine Thrust, N** \\ \hline \(x_{v}\) & 1500 & 150 \\ \hline \(y_{v}\) & 1700 & 75 \\ \hline \(z_{v}\) & 1800 & 75 \\ \hline \(x_{v}y_{v}\) & 50 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the spacecraft and engines (test version).