A New Concept of UAV Recovering System

Abstract

This paper introduces a new concept of recovering UAVs to their carrier using manipulator, aiming to better use the carrier’s carrying capability. The state-of-the-art of recycling UAVs are stated in the first place, the advantages and the setbacks of the current available recycling systems are introduced, and the reason for the setbacks are further analyzed before this new concept is introduced. To accomplish the basic recovering task, a minimum system configuration that can explain the idea is introduced and the model of the system is built based on several important assumptions. To further explain how the system works, a simulation based on the above configuration is given. The simulation result indicates that the system can fulfill the task of recovering the UAV, and the result also verified the efficiency of the new system, which implies its potential usefulness in the future application.

Appendices

Appendix 1

$$ \left( {\begin{array}{*{20}c} {\dot{x}_{1} } \\ {\dot{x}_{2} } \\ {\dot{x}_{3} } \\ {\dot{x}_{4} } \\ {\dot{x}_{5} } \\ {\dot{x}_{6} } \\ {\dot{x}_{7} } \\ {\dot{x}_{8} } \\ {\dot{x}_{9} } \\ {\dot{x}_{10} } \\ {\dot{x}_{11} } \\ {\dot{x}_{12} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {x_{4} + x_{5} \sin x_{1} \tan x_{2} + x_{6} \cos x_{1} \tan x_{2} } \\ {x_{5} \cos x_{1} - x_{6} \sin x_{1} } \\ {\left( {x_{5} \sin x_{1} + x_{6} \cos x_{1} } \right)/\cos x_{2} } \\ {\left( {\left( {I_{yy} - I_{zz} } \right)x_{5} x_{6} - x_{5}\Omega Jr} \right)/I_{xx} } \\ {\left( {\left( {I_{zz} - I_{xx} } \right)x_{4} x_{6} + x_{4}\Omega Jr} \right)/I_{yy} } \\ {\left( {\left( {I_{xx} - I_{yy} } \right)x_{4} x_{5} - {\dot{\Omega }}Jr} \right)/I_{zz} } \\ {x_{10} } \\ {x_{11} } \\ {x_{12} } \\ 0 \\ 0 \\ g \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {1/I_{xx} } & 0 & 0 \\ 0 & 0 & {1/I_{yy} } & 0 \\ 0 & 0 & 0 & {1/I_{zz} } \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{{\cos x_{1} \sin x_{2} \cos x_{3} + \sin x_{1} \sin x_{3} }}{m}} & 0 & 0 & 0 \\ {\frac{{\cos x_{1} \sin x_{2} \sin x_{3} - \sin x_{1} \cos x_{3} }}{m}} & 0 & 0 & 0 \\ { - \frac{{\cos x_{1} \cos x_{2} }}{m}} & 0 & 0 & 0 \\ \end{array} } \right)\left( {\begin{array}{*{20}c} F \\ {\tau_{1q} } \\ {\tau_{2q} } \\ {\tau_{3q} } \\ \end{array} } \right) $$

Appendix 2

$$ \begin{aligned} & \varvec{M} = \left[ {\begin{array}{*{20}c} {m_{11} } & {m_{12} } & {m_{13} } \\ {m_{21} } & {m_{22} } & {m_{23} } \\ {m_{31} } & {m_{32} } & {m_{33} } \\ \end{array} } \right] \\ & \quad \quad m_{11} = \frac{{{\text{I}}_{2xx} + {\text{I}}_{3xx} + {\text{I}}_{2yy} }}{2} + {\text{I}}_{3yy} + {\text{I}}_{1zz} - \frac{{\left( {{\text{I}}_{3xx} + {\text{I}}_{3yy} + {\text{m}}_{3} {\text{r}}_{3com}^{2} } \right){\text{cos(2}}q_{2} + 2q_{3} )}}{2} \\ & \quad \quad \quad \,\, + \frac{{{\text{l}}_{2}^{2} {\text{m}}_{3} + {\text{m}}_{2} {\text{r}}_{2com}^{2} + {\text{m}}_{3} {\text{r}}_{3com}^{2} }}{2} + \frac{{\left( {{\text{I}}_{2yy} - {\text{I}}_{2xx} {\text{ + l}}_{2}^{2} {\text{m}}_{3} + {\text{ m}}_{2} {\text{r}}_{2com}^{2} } \right){\text{cos2}}q_{2} }}{2} \\ & \quad \quad \quad \,\, + {\text{l}}_{2} {\text{m}}_{3} {\text{r}}_{3com} { \cos }q_{3} + {\text{l}}_{2} {\text{m}}_{3} {\text{r}}_{3com} {\text{cos(2}}q_{2} + q_{3} )\\ & \quad \quad m_{12} = 0 \\ & \quad \quad m_{13} = 0 \\ & \quad \quad m_{22} = {\text{m}}_{3} {\text{l}}_{2}^{2} + 2 {\text{m}}_{3} { \cos }q_{3} {\text{l}}_{2} {\text{r}}_{3com} + {\text{m}}_{2} {\text{r}}_{2com}^{2} + {\text{m}}_{3} {\text{r}}_{3com}^{2} + {\text{I}}_{2zz} + {\text{I}}_{3zz} \\ & \quad \quad m_{23} = m_{32} = {\text{m}}_{3} {\text{r}}_{3com}^{2} + {\text{l}}_{2} {\text{m}}_{3} { \cos }q_{3} {\text{r}}_{3com} + {\text{I}}_{3zz} \\ & \quad \quad m_{33} = {\text{m}}_{3} {\text{r}}_{3com}^{2} + {\text{I}}_{3zz} \\ \end{aligned} $$

$$ \begin{aligned} & \varvec{C}\left( {\varvec{q},\dot{\varvec{q}}} \right) = \\ & \quad \quad \left[ \begin{aligned} & - \dot{q}_{1} (\left( {{\text{I}}_{2yy} \dot{q}_{2} - {\text{I}}_{2xx} \dot{q}_{2} + {\text{l}}_{2}^{2} {\text{m}}_{ 3} \dot{q}_{2} + {\text{m}}_{ 2} \dot{q}_{2} r_{2com}^{2} } \right){\text{sin2}}q_{2} \\ & + \left( {{\text{I}}_{3yy} \dot{q}_{3} + {\text{ I}}_{3yy} \dot{q}_{2} - {\text{I}}_{3xx} \dot{q}_{3} - {\text{I}}_{3xx} \dot{q}_{2} + {\text{m}}_{ 3} \dot{q}_{2} r_{3com}^{2} + {\text{m}}_{ 3} \dot{q}_{3} r_{3com}^{2} } \right){\text{sin(2}}q_{2} + 2q_{3} ) { } \\ & + \left( { 2 {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{2} r_{3com} + {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{2} r_{3com} } \right){\text{sin(2}}q_{2} + q_{3} )+ {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{3} r_{3com} { \sin }q_{3} )\\ & \quad (\left( {{\text{I}}_{2yy} \dot{q}_{1}^{2} - {\text{I}}_{2xx} \dot{q}_{1}^{2} } \right){\text{sin2}}q_{2} ) / 2- (\left( {{\text{I}}_{3xx} \dot{q}_{1}^{2} + {\text{I}}_{3yy} \dot{q}_{1}^{2} } \right){\text{sin(2}}q_{2} + 2q_{3} ) ) / 2 { } \\ & \quad + (\left( {{\text{l}}_{2}^{2} {\text{m}}_{ 3} \dot{q}_{1}^{2} + {\text{m}}_{ 2} \dot{q}_{1}^{2} r_{2com}^{2} } \right){\text{sin2}}q_{2} ) / 2+ ( {\text{m}}_{ 3} \dot{q}_{1}^{2} r_{3com}^{2} {\text{sin(2}}q_{2} + 2q_{3} ) ) / 2 { } \\ & \quad - {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{3}^{2} r_{3com} { \sin }q_{3} + {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{1}^{2} r_{3com} {\text{sin(2}}q_{2} + q_{3} )- 2 {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{2} \dot{q}_{3} r_{3com} { \sin }q_{3} \\ & \quad \quad \quad ( {\text{I}}_{3yy} \dot{q}_{1}^{2} {\text{sin(2}}q_{2} + 2q_{3} ) ) / 2- ( {\text{I}}_{3xx} \dot{q}_{1}^{2} {\text{sin(2}}q_{2} + 2q_{3} ) ) / 2 { } \\ & \quad \quad \quad + ( {\text{m}}_{ 3} \dot{q}_{1}^{2} r_{3com}^{2} {\text{sin(2}}q_{2} + 2q_{3} ) ) / 2+ ( {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{1}^{2} r_{3com} { \sin }q_{3} ) / 2 { } \\ & \quad \quad \quad + {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{2}^{2} r_{3com} { \sin }q_{3} + ( {\text{l}}_{ 2} {\text{m}}_{ 3} \dot{q}_{1}^{2} r_{3com} {\text{sin(2}}q_{2} + q_{3} ) ) / 2\\ \end{aligned} \right] \\ \end{aligned} $$

$$ \begin{aligned} & \varvec{G}\left( \varvec{q} \right) = \\ & \quad \left[ {\begin{array}{*{20}c} 0 \\ { - {\text{m}}_{ 3} {\text{g(}}r_{3com} ( {\text{cos}}q_{2} { \cos }q_{3} - { \sin }q_{2} { \sin }q_{3} )+ {\text{l}}_{ 2} { \cos }q_{2} )- {\text{m}}_{ 2} {\text{g}}r_{2com} { \cos }q_{2} } \\ { - {\text{m}}_{ 3} {\text{g}}r_{3com} ( {\text{cos}}q_{2} { \cos }q_{3} - { \sin }q_{2} { \sin }q_{3} )} \\ \end{array} } \right] \\ \end{aligned} $$

Appendix 3

$$ \begin{aligned} & q_{1} = a\tan 2\left( {x_{y} ,x_{x} } \right) \\ & q_{3} = \pm \left( {\pi - a\cos \left( {\frac{{\left( {x_{z} - l_{1} } \right)^{2} + x_{x}^{2} + x_{y}^{2} - l_{2}^{2} - l_{3}^{2} }}{{2l_{2} l_{3} }}} \right)} \right) \\ & q_{2} = a\tan 2\left( {x_{z} - l_{1} ,\sqrt {x_{x}^{2} + x_{y}^{2} } } \right) - a\tan 2\left( {l_{2} + l_{3} \cos q_{3} ,l_{3} \sin q_{3} } \right) \\ \end{aligned} $$