# A Robust Control Algorithm of a Descent Vehicle Angular Motion in the Earth's Atmosphere
## 4 Standard Algorithm and Conditions for Its Comparison with the New Algorithm  
The standard motion control of a capsule-type lander from a constant thrust provided by reaction engine is formed in two stages. First, the vector of control signals is found  
\[\mathbf{u}=\mathbf{u}_{0}-\mathbf{F}\Delta\mathbf{y}_{att}, \tag{5}\]  
where \(\mathbf{u}_{0}\) - feedforward control to control, \(\mathbf{F}\) - output regulator matrix, \(\Delta\mathbf{y}_{attt}=\mathbf{y}_{att}-\mathbf{y}_{att,pr}\), and \(\mathbf{y}_{att,pr}\) - the programmed value of the output vector (2), consisting of the programmed values of the inertial roll angle \(\tilde{\gamma}_{pr}\) and the vector of the angular velocity of the associated CS \(\mathbf{\omega}^{E_{v}}_{E_{v},pr}\). Then the signals (5) are converted into the control torque of constant thrust of reaction engine (the duration of switching on the reaction engine) according to the piecewise linear law with a dead zone and saturation.  
The standard stabilization algorithm is empirical. In it, control (5) turns out to be autonomous in the roll (\(\gamma\), \(\omega^{E_{v}}_{x_{v}}\)), yaw (\(\beta\), \(\omega^{E_{v}}_{y_{v}}\)) and pitch (\(\alpha\), \(\omega^{E_{v}}_{z_{v}}\)) channels. In the atmospheric section, it is aimed at damping the angular velocities \(\mathbf{\omega}^{E_{v}}_{E_{v}}\) and tracking the programmed roll angle (\(\gamma\rightarrow\gamma_{pr}\)). The balancing position (\(\alpha\rightarrow\alpha_{balt}\), \(\beta\rightarrow\beta_{bal}\)) is maintained due to the static stability of the lander [23].  
In the simplest version of law (5)  
\[\mathbf{u}_{0}=\mathbf{\omega}^{E_{v}}_{E_{v},pr}=\mathbf{0}_{3\times 1},\quad\tilde{\gamma}_{ pr}=\gamma_{pr}.\]Hereinafter, \(\mathbf{0}_{n\times m}\) is the zero matrix of dimension \(n\times m\). To improve the orientation accuracy and create the same conditions when comparing the standard and new algorithms (according to the influence of the regulator matrices), it is advisable to use the conversion of the speed roll angle into the inertial roll angle  
\[\tilde{\gamma}_{pr}=\tilde{\gamma}\Big{(}t,\mathbf{x}_{trj},\mathbf{\theta}_{pr}^{E+Q_{ v}}\Big{)}\]  
and taking into account the angular velocity of the reference CS \(Q_{v}\)  
\[\mathbf{\omega}_{E_{v},pr}^{E_{v}}=\mathbf{\omega}_{E_{v}}^{Q_{v}}\Big{(}\mathbf{x}_{trj}, \mathbf{\theta}_{pr}^{E+Q_{v}}\Big{)}.\]  
Since in model (4) in the balancing position  
\[\mathbf{M}_{E_{v}}^{net}\Big{(}\mathbf{x}_{trj},\mathbf{\theta}_{pr}^{E+Q_{v}}\Big{)}\equiv \mathbf{0}_{3\times 1},\]  
the feedforward control without taking into account the current kinematics of the lander motion is  
\[\mathbf{u}_{-0}=-\mathbf{M}_{E_{v}}^{Syr}\Big{(}\mathbf{\omega}_{E_{v},pr}^{E_{v}}\Big{)}. \tag{6}\]  
In real time, the angular velocity \(\mathbf{\omega}_{E_{v}}^{Q_{v}}\), which depends, in particular, on the aerodynamic force (a strongly varying stochastic vector), is difficult to calculate. But as the programmed angular velocity with some approximation, you can use its part  
\[\mathbf{\omega}_{E_{v},pr}^{E_{v}}=\mathbf{\omega}_{E_{v}}^{Q_{grn}}\Big{(}\mathbf{x}_{trj },\mathbf{\theta}_{pr}^{E+Q_{v}}\Big{)}, \tag{7}\]  
i.e., take into account only the angular velocity of the Earth's rotation around its axis.