PSO-based online estimation of aerodynamic ground effect in the backstepping controller of quadrotor

Abstract

Stability of landing and forward flight of unmanned aerial vehicles near the ground is one of the challenging problems in the quadrotor control system. In real-time operation, when the rotary-wing vehicle flies close to the ground, the thrust increases are called ground effect. According to the flight and environmental conditions and the quality of ground surface, the aerodynamic ground effect is varied continuously. In this paper, a backstepping controller is designed and adjusted by using particle swarm optimization. Controller adjustment is referred to as the process of online estimation of the ground effect in various conditions. The stability analysis is performed and guaranteed by Lyapunov theory. The simulation results indicate that the proposed method is highly practical and capable of tracking the desired trajectory appropriately.

Appendix

The body of quadrotor is assumed symmetry, and the center of mass is coincided the center of geometry. Then the inertia matrix ($I$) is diagonal: $I = {\text{diag}}\left( {I_{xx} , I_{yy} , I_{zz} } \right)$ (N m s²). The total thrust and moments produced by four rotors are extracted by:

$$ U_{B} \left(\Omega \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ {U_{1} } \\ \end{array} } \\ {U_{2} } \\ {\begin{array}{*{20}c} {U_{3} } \\ {U_{4} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ {b\left( {\Omega _{1}^{2} +\Omega _{2}^{2} +\Omega _{3}^{2} +\Omega _{4}^{2} } \right)} \\ {\begin{array}{*{20}c} {bl\left( {\Omega _{4}^{2} -\Omega _{2}^{2} } \right)} \\ {bl\left( {\Omega _{3}^{2} -\Omega _{1}^{2} } \right)} \\ {d\left( {\Omega _{2}^{2} +\Omega _{4}^{2} -\Omega _{1}^{2} -\Omega _{3}^{2} } \right)} \\ \end{array} } \\ \end{array} } \right] $$

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where $b\left( {{\text{N}}\;{\text{s}}^{2} } \right)$ indicates the thrust factor, $d\left( {{\text{N}}\,{\text{m}}\,{\text{s}}^{2} } \right)$ is drag factor, $l\left( {\text{m}} \right)$ is the distance between the centers of the quadrotor and the rotor, $U_{i}$ is the input vector components, and $\Omega _{i}$ is the speed of rotor. Refer [18] for more information on the assumptions and saturation limits of control input vector. According to Sect. 3, the dynamic equations of quadrotor under the ground effect are reformulated as follows:

$$ \left[ {\begin{array}{*{20}l} {\dot{x}_{1} } \hfill \\{\dot{x}_{2} } \hfill \\ {\dot{x}_{3} } \hfill \\ {\dot{x}_{4} }\hfill \\ {\dot{x}_{5} } \hfill \\ {\dot{x}_{6} } \hfill \\{\dot{x}_{7} } \hfill \\ {\dot{x}_{8} } \hfill \\ {\dot{x}_{9} }\hfill \\ {\dot{x}_{10} } \hfill \\ {\dot{x}_{11} } \hfill \\{\dot{x}_{12} } \hfill \\ \end{array} } \right] = \left[{\begin{array}{*{20}l} {\dot{\varphi }} \hfill \\ {\ddot{\varphi }}\hfill \\ {\dot{\theta }} \hfill \\ {\ddot{\theta }} \hfill \\ {\dot{\psi }} \hfill \\ {\ddot{\psi }} \hfill \\ {\dot{x}} \hfill \\{\ddot{x}} \hfill \\ {\dot{y}} \hfill \\ {\ddot{y}} \hfill \\{\dot{z}} \hfill \\ {\ddot{z}} \hfill \\ \end{array} } \right] =\left[ \begin{array}{c}x_2\\ x_{4} x_{6} a_{1} - \frac{{J_{TP} }}{{I_{xx}}}\dot{\theta }\Omega + \frac{{b_{1} }}{{1 - \left({\frac{R}{{4x_{11} }}} \right)^{2} \left\{ {\frac{1}{{1 + \left({\frac{{\sqrt {x_{8}^{2} + x_{10}^{2} + x_{12}^{2} } }}{{\sqrt { -\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2} + \sqrt {\left({\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2}} \right)^{2} +\left( {\frac{{b\Omega _{i}^{2} }}{2\rho A}} \right)^{2} } } }}}\right)^{2} }}} \right\}}}\left( {U_{2} } \right) \hfill\\x_4 \\ x_{2}x_{6} a_{3} - \frac{{J_{TP} }}{{I_{xx} }}\dot{\varphi }\Omega +\frac{{b_{2} }}{{1 - \left( {\frac{R}{{4x_{11} }}} \right)^{2}\left\{ {\frac{1}{{1 + \left( {\frac{{\sqrt {x_{8}^{2} + x_{10}^{2}+ x_{12}^{2} } }}{{\sqrt { - \frac{{x_{8}^{2} + x_{10}^{2} +x_{12}^{2} }}{2} + \sqrt {\left( {\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2}} \right)^{2} + \left( {\frac{{b\Omega _{i}^{2}}}{2\rho A}} \right)^{2} } } }}} \right)^{2} }}} \right\}}}\left({U_{3} } \right) \hfill\\x_6\\ x_{4} x_{2} a_{5} + b_{3} (U_{4} ) \\x_8 \\ \frac{{u_{x} /m}}{{1 - \left({\frac{R}{{4x_{11} }}} \right)^{2} \left\{ {\frac{1}{{1 + \left({\frac{{\sqrt {x_{8}^{2} + x_{10}^{2} + x_{12}^{2} } }}{{\sqrt { -\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2} + \sqrt {\left({\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2}} \right)^{2} + \left( {\frac{{b\Omega _{i}^{2} }}{2\rho A}} \right)^{2} } } }}}\right)^{2} }}} \right\}}}\left( {U_{1} } \right) - K_{x} x_{8} /m\hfill\\10 \\ \frac{{u_{y} /m}}{{1 - \left( {\frac{R}{{4x_{11} }}}\right)^{2} \left\{ {\frac{1}{{1 + \left( {\frac{{\sqrt {x_{8}^{2} +x_{10}^{2} + x_{12}^{2} } }}{{\sqrt { - \frac{{x_{8}^{2} +x_{10}^{2} + x_{12}^{2} }}{2} + \sqrt {\left( {\frac{{x_{8}^{2} +x_{10}^{2} + x_{12}^{2} }}{2}} \right)^{2} + \left( {\frac{{b\Omega_{i}^{2} }}{2\rho A}} \right)^{2} } } }}} \right)^{2} }}}\right\}}}\left( {U_{1} } \right) - K_{y} x_{10} /m \hfill\\x_{12} \\ \frac{{\left( {\cos x_{1} \cos x_{3} } \right)/m}}{{1 - \left({\frac{R}{{4x_{11} }}} \right)^{2} \left\{ {\frac{1}{{1 + \left({\frac{{\sqrt {x_{8}^{2} + x_{10}^{2} + x_{12}^{2} } }}{{\sqrt { -\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2} + \sqrt {\left({\frac{{x_{8}^{2} + x_{10}^{2} + x_{12}^{2} }}{2}} \right)^{2} +\left( {\frac{{b\Omega _{i}^{2} }}{2\rho A}} \right)^{2} } } }}}\right)^{2} }}} \right\}}}\left( {U_{1} } \right) - K_{z} x_{12} /m\hfill \\ \end{array} \right] $$

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$$ \begin{aligned} a_{1} & = \left( {I_{yy} - I_{zz} } \right)/I_{xx} ,\quad a_{3} = \left( {I_{zz} - I_{xx} } \right)/I_{yy} ,\quad a_{1} = \left( {I_{xx} - I_{yy} } \right)/I_{zz} \\ b_{1} & = d/I_{xx} ,\quad b_{2} = d/I_{yy} ,\quad b_{3} = 1/I_{zz} \\ \end{aligned} $$